License
-------
The `aclib' package is licensed under the Artistic License 2.0.
aclib-1.3.2/init.g 0000664 0003717 0003717 00000000454 13627151544 015127 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W init.g share package Karel Dekimpe
#W Bettina Eick
##
# read .gd files
ReadPackage( "aclib", "gap/groups.gd" );
aclib-1.3.2/read.g 0000664 0003717 0003717 00000001714 13627151544 015077 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W read.g share package Karel Dekimpe
#W Bettina Eick
##
# read matrix groups
ReadPackage( "aclib", "gap/matgrp3.gi" );
ReadPackage( "aclib", "gap/matgrp4.gi" );
ReadPackage( "aclib", "gap/matgrp.gi" );
# read corresponding pcp groups
ReadPackage( "aclib", "gap/pcpgrp3.gi" );
ReadPackage( "aclib", "gap/pcpgrp4.gi" );
ReadPackage( "aclib", "gap/pcpgrp.gi" );
ReadPackage( "aclib", "gap/betti.gi" );
ReadPackage( "aclib", "gap/union.gi" );
ReadPackage( "aclib", "gap/extend.gi" );
# read some help functions to work with crystallographic groups
if IsPackageMarkedForLoading( "crystcat", "1.1" ) then
ReadPackage( "aclib", "gap/crystgrp.gi" );
else
Print("#I The crystcat package is not installed \n");
Print("#I Cannot load crystallographic groups functionality \n");
fi;
aclib-1.3.2/htm/chapters.htm 0000664 0003717 0003717 00000001405 13627151544 017124 0 ustar gap-jenkins gap-jenkins aclib : a GAP 4 package - Chapters
aclib : a GAP 4 package - Chapters
- The Almost Crystallographic Groups Package
- Algorithms for almost crystallographic groups
- The catalog of almost crystallographic groups
- Example computations with almost crystallographic groups
aclib manual
January 2020
aclib-1.3.2/htm/biblio.htm 0000664 0003717 0003717 00000003245 13627151544 016557 0 ustar gap-jenkins gap-jenkins aclib : a GAP 4 package - References
aclib : a GAP 4 package - References
- [AUS]
-
Louis Auslander.
Bieberbach's theorem on space groups and discrete uniform subgroups
of lie groups.
Ann. of Math, 71(3):579--590, 1960.
- [BRO]
-
Kenneth S. Brown.
Cohomology of Groups, volume 87 of Grad. Texts in Math.
Springer-Verlag, New York, 1982.
- [DE2]
-
Karel Dekimpe and Bettina Eick.
Computational aspects of group extensions and their application in
topology.
Submitted, 2001.
- [DE1]
-
Karel Dekimpe and Bettina Eick.
Computations with almost crystallographic groups.
Submitted, 2001.
- [KD]
-
Karel Dekimpe.
Almost-Bieberbach Groups: Affine and Polynomial Structures,
volume 1639 of Lecture Notes in Mathematics.
Springer-Verlag, Heidelberg, 1996.
- [DM]
-
An Descheemaeker and Wim Malfait.
P-localization of relative groups.
Journal of Pure and Applied Algebra, 159(1), 2001.
- [LEE]
-
Kyung Bai Lee.
There are only finitely many infra-nilmanifolds under each
nilmanifold.
Quart. J. Math. Oxford, 39(2):61--66, 1988.
[Up]
aclib manual
January 2020
aclib-1.3.2/htm/CHAP004.htm 0000664 0003717 0003717 00000030634 13627151544 016260 0 ustar gap-jenkins gap-jenkins [aclib] 4 Example computations with almost crystallographic groups
[Up] [Previous] [Index]
4 Example computations with almost crystallographic groups
Sections
- Example computations I
- Example computations II
- Example computations III
Using the functions available for pcp groups in the share package
polycyclic it is now easy to redo some of the calculations of
KD. As a first example we check whether the groups indicated
as torsion free in KD are also determined as torsion free
ones by GAP. In KD these almost Bieberbach groups are listed as
``AB-groups''. So for type ``013'' these are the groups with parameters
(k,0,1,0,1,0) where k is an even integer. Let's look at some examples
in GAP:
gap> G:=AlmostCrystallographicPcpDim4("013",[8,0,1,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
true
gap> G:=AlmostCrystallographicPcpDim4("013",[9,0,1,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
false
Further, there is also some cohomology information in the tables
of KD. In fact, the groups in this library were obtained
as extensions E of the form
where, in the 4-dimensional case Q = E/〈d 〉. The
cohomology information for the particular example above shows that
the groups determined by a parameter set (k1,k2,k3,k4,k4,k6)
are equivalent as extensions to the groups determined by the parameters
(k1, k2 mod 2, k3 mod 2, k4 mod 2, k5 mod 2, 0). This is
also visible in finding torsion:
gap> G:=AlmostCrystallographicPcpDim4("013",[10,0,2,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
false
gap> G:=AlmostCrystallographicPcpDim4("013",[10,0,3,0,1,9]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
true
The computation of cohomology groups played an important role in the
classification of the almost Bieberbach groups in KD. Using
GAP, it is now possible to check these computations. As an example we
consider the 4-dimensional almost crystallographic groups of type 85 on
page 202 of KD. This group E has 6 generators. In the table, one
also finds the information
for Q=E/〈d 〉 as above. Moreover, the Q--module Z is
in fact the group 〈d 〉, where the Q-action comes from
conjugation inside E. In the case of groups of type 85, Z is a
trivial Q-module. The following example demonstrates how to (re)compute
this two-cohomology group H2(Q,Z).
gap> G:=AlmostCrystallographicPcpGroup(4, "085", false);
Pcp group with orders [ 2, 4, 0, 0, 0, 0 ]
gap> GroupGeneratedByd:=Subgroup(G, [G.6] );
Pcp group with orders [ 0 ]
gap> Q:=G/GroupGeneratedByd;
Pcp group with orders [ 2, 4, 0, 0, 0 ]
gap> action:=List( Pcp(Q), x -> [[1]] );
[ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ]
gap> C:=CRRecordByMats( Q, action);;
gap> TwoCohomologyCR( C ).factor.rels;
[ 2, 2, 4, 0 ]
This last line gives us the abelian invariants of the second
cohomology group H2(Q,Z). So we should read this line as
which indeed coincides with the information in KD.
As another application of the capabilities of the combination of
aclib and polycyclic we check some computations of DM.
Section 5 of the paper DM is completely devoted to an example
of the computation of the P-localization of a virtually nilpotent group,
where P is a set of primes. Although it is not our intention to
develop the theory of P-localization of groups at this place, let us
summarize some of the main results concerning this topic here.
For a set of primes P, we say that n ∈ P if and only if n is
a product of primes in P. A group G is said to be P-local if and
only if the map μn:G→ G: g → gn is bijective for
all n ∈ P′, where P′ is the set of all primes not in P. The
P-localization of a group G, is a P-local group GP together
with a morphism α:G → GP which satisfy the following
universal property: For each P-local group L and any morphism
φ: G → L, there exists a unique morphism ψ:GP → L, such that ψ°α = φ.
This concept of localization is well developed for finite groups and
for nilpotent groups. For a finite group G, the P-localization is
the largest quotient of G, having no elements with an order belonging to
P′ (the morphism α, mentioned above is the natural projection).
In DM a contribution is made towards the localization of virtually
nilpotent groups. The theory developed in the paper is then illustrated
in the last section of the paper by means of the computation of the
P-localization of an almost crystallographic group. For their example
the authors have chosen an almost crystallographic group G of dimension 3
and type 17. For the set of parameters (k1,k2,k3,k4) they have
considered all cases of the form (k1,k2,k3,k4)=(2,0,0,k4).
Here we will check their computations in two cases k4=0 and k4=1
using the set of primes P={2}. The holonomy group of these almost
crystallographic groups G is the dihedral group D6 of order
12. Thus there is a short exact sequence of the form
| 1 → Fitt(G) → G → D6 → 1· |
|
As a first step in their computation, Descheemaeker and Malfait determine
the group IP′D6, which is the unique subgroup of order 3 in
D6. One of the main objects in DM is the group K=p−1 (IP′D6), where p is the natural projection of G onto its
holonomy group. It is known that the P-localization of G coincides
with the P-localization of G/γ3(K), where γ3(K) is the
third term in the lower central series of K. As G/γ3(K) is
finite in this example, we exactly know what this P-localization is.
Let us now show, how GAP can be used to compute this P-localization in
two cases:
First case: The parameters are (k1,k2,k3,k4)=(2,0,0,0)
gap> G := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,0] );
Pcp group with orders [ 2, 6, 0, 0, 0 ]
gap> projection := NaturalHomomorphismOnHolonomyGroup( G );
[ g1, g2, g3, g4, g5 ] -> [ g1, g2, identity, identity, identity ]
gap> F := HolonomyGroup( G );
Pcp group with orders [ 2, 6 ]
gap> IPprimeD6 := Subgroup( F , [F.2^2] );
Pcp group with orders [ 3 ]
gap> K := PreImage( projection, IPprimeD6 );
Pcp group with orders [ 3, 0, 0, 0 ]
gap> PrintPcpPresentation( K );
pcp presentation on generators [ g2^2, g3, g4, g5 ]
g2^2 ^ 3 = identity
g3 ^ g2^2 = g3^-1*g4^-1
g3 ^ g2^2^-1 = g4*g5^-2
g4 ^ g2^2 = g3*g5^2
g4 ^ g2^2^-1 = g3^-1*g4^-1*g5^2
g4 ^ g3 = g4*g5^2
g4 ^ g3^-1 = g4*g5^-2
gap> Gamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));
Pcp group with orders [ 0, 0, 0 ]
gap> quotient := G/Gamma3K;
Pcp group with orders [ 2, 6, 3, 3, 2 ]
gap> S := SylowSubgroup( quotient, 3);
Pcp group with orders [ 3, 3, 3 ]
gap> N := NormalClosure( quotient, S);
Pcp group with orders [ 3, 3, 3 ]
gap> localization := quotient/N;
Pcp group with orders [ 2, 2, 2 ]
gap> PrintPcpPresentation( localization );
pcp presentation on generators [ g1, g2, g3 ]
g1 ^ 2 = identity
g2 ^ 2 = identity
g3 ^ 2 = identity
This shows that GP ≅ Z23.
Second case: The parameters are (k1,k2,k3,k4)=(2,0,0,1)
gap> G := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,1]);;
gap> projection := NaturalHomomorphismOnHolonomyGroup( G );;
gap> F := HolonomyGroup( G );;
gap> IPprimeD6 := Subgroup( F , [F.2^2] );;
gap> K := PreImage( projection, IPprimeD6 );;
gap> Gamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));;
gap> quotient := G/Gamma3K;;
gap> S := SylowSubgroup( quotient, 3);;
gap> N := NormalClosure( quotient, S);;
gap> localization := quotient/N;
Pcp group with orders [ 2, 2, 2 ]
gap> PrintPcpPresentation( localization );
pcp presentation on generators [ g1, g2, g3 ]
g1 ^ 2 = identity
g2 ^ 2 = g3
g3 ^ 2 = identity
g2 ^ g1 = g2*g3
g2 ^ g1^-1 = g2*g3
In this case, we see that GP=D4.
The reader can check that these results coincide with those obtained in
DM. Note also that we used a somewhat different scheme to compute
this localization than the one used in DM. We invite the reader to
check the same computations, tracing exactly the steps made in DM.
[Up] [Previous] [Index]
aclib manual
January 2020
aclib-1.3.2/htm/theindex.htm 0000664 0003717 0003717 00000006545 13627151544 017135 0 ustar gap-jenkins gap-jenkins aclib : a GAP 4 package - Index
aclib : a GAP 4 package - Index
A
B
D
E
H
I
M
N
O
P
R
T
- Algorithms for almost crystallographic groups 2.0
- AlmostCrystallographicDim3 3.1.1
- AlmostCrystallographicDim4 3.1.1
- AlmostCrystallographicGroup 3.1.1
- AlmostCrystallographicInfo 3.3.1
- AlmostCrystallographicPcpDim3 3.2.1
- AlmostCrystallographicPcpDim4 3.2.1
- AlmostCrystallographicPcpGroup 3.2.1
- Betti numbers 2.2
- BettiNumber 2.2.2
- BettiNumbers 2.2.3
- Determination of certain extensions 2.3
- Example computations I 4.1
- Example computations II 4.2
- Example computations III 4.3
- Example computations with almost crystallographic groups 4.0
- HasExtensionOfType 2.3.1
- HolonomyGroup 3.2.3
- IsAlmostBieberbachGroup 2.1.2
- IsAlmostCrystallographic 2.1.1
- IsomorphismPcpGroup 3.2.2
- More about almost crystallographic groups 1.1
- More about the type and the defining parameters 3.3
- NaturalHomomorphismOnHolonomyGroup 3.2.3
- OrientationModule 2.2.1
- Polycyclically presented groups 3.2
- Properties of almost crystallographic groups 2.1
- Rational matrix groups 3.1
- The Almost Crystallographic Groups Package 1.0
- The catalog of almost crystallographic groups 3.0
- The electronic versus the printed library 3.4
[Up]
aclib manual
January 2020
aclib-1.3.2/htm/CHAP001.htm 0000664 0003717 0003717 00000015351 13627151544 016254 0 ustar gap-jenkins gap-jenkins [aclib] 1 The Almost Crystallographic Groups Package
[Up] [Next] [Index]
1 The Almost Crystallographic Groups Package
Sections
- More about almost crystallographic groups
A group is called almost crystallographic if it is a finitely generated
nilpotent-by-finite group without non-trivial finite normal subgroups. An
important special case of almost crystallographic groups are the almost
Bieberbach groups: these are almost crystallographic and torsion free.
By its definition, an almost crystallographic group G has a finitely
generated nilpotent normal subgroup N of finite index. Clearly, N is
polycyclic and thus has a polycyclic series. The number of infinite cyclic
factors in such a series for N is an invariant of G: the Hirsch length
of G.
For each almost crystallographic group of Hirsch length 3 and 4 there exists
a representation as a rational matrix group in dimension 4 or 5, respectively.
These representations can be considered as affine representations of dimension
3 or 4. Via these representations, the almost crystallographic groups act
(properly discontinuously) on R3 or R4. That is one reason to define
the dimension of an almost crystallographic group as its Hirsch length.
The 3-dimensional and a part of the 4-dimensional almost crystallographic
groups have been classified by K. Dekimpe in KD. This classification
includes all almost Bieberbach groups in dimension 3 and 4. It is the first
central aim of this package to give access to the resulting library of groups.
The groups in this electronic catalog are available in two different
representations: as rational matrix groups and as polycyclically presented
groups. While the first representation is the more natural one, the latter
description facilitates effective computations with the considered groups
using the methods of the Polycyclic package.
The second aim of this package is to introduce a variety of algorithms for
computations with polycyclically presented almost crystallographic groups.
These algorithms supplement the methods available in the Polycyclic
package and give access to some methods which are interesting specifically
for almost crystallographic groups. In particular, we present methods to
compute Betti numbers and to construct or check the existence of certain
extensions of almost crystallographic groups. We note that these methods
have been applied in DE1 and DE2 for computations with
almost crystallographic groups.
Finally, we remark that almost crystallographic groups can be seen as natural
generalizations of crystallographic groups. A library of crystallographic
groups and algorithms to compute with crystallographic groups are available
in the GAP packages cryst, carat and crystcat.
Almost crystallographic groups were first discussed in the theory of
actions on Lie groups. We recall the original definition here briefly
and we refer to AUS, KD and LEE for more details.
Let L be a connected and simply connected nilpotent Lie group. For
example, the 3-dimensional Heisenberg group, consisting of all upper
unitriangular 3×3--matrices with real entries is of this type.
Then L\rtimes Aut(L) acts affinely (on the left) on L via
| ∀l,l′ ∈ L,∀α ∈ Aut(L): (l,α)l′=l α(l′)· |
|
Let C be a maximal compact subgroup of Aut(L). Then a subgroup G
of L \rtimes C is said to be an almost crystallographic group if and only
if the action of G on L, induced by the action of L\rtimes Aut(L),
is properly discontinuous and the quotient space G \L is compact.
One recovers the situation of the ordinary crystallographic groups by taking
L=\Bbb Rn, for some n, and C=O(n), the orthogonal group.
More generally, we say that an abstract group is an almost crystallographic
group if it can be realized as a genuine almost crystallographic subgroup
of some L \rtimes C. In the following theorem we outline some algebraic
characterizations of almost crystallographic groups; see Theorem 3.1.3 of
KD. Recall that the Fitting subgroup Fitt(G) of a
polycyclic-by-finite group G is its unique maximal normal nilpotent
subgroup.
Theorem.
The following are equivalent for a polycyclic-by-finite group G:
- (1)
- G is an almost crystallographic group.
- (2)
- Fitt(G) is torsion free and of finite index in G.
- (3)
- G contains a torsion free nilpotent normal subgroup N
of finite index in G with CG(N) torsion free.
- (4)
- G has a nilpotent subgroup of finite index and there
are no non-trivial finite normal subgroups in G.
In particular, if G is almost crystallographic, then G / Fitt(G)
is finite. This factor is called the holonomy group of G.
The dimension of an almost crystallographic group equals the dimension
of the Lie group L above which coincides also with the Hirsch length
of the polycyclic-by-finite group. This library therefore contains
families of virtually nilpotent groups of Hirsch length 3 and 4.
[Up] [Next] [Index]
aclib manual
January 2020
aclib-1.3.2/htm/CHAP003.htm 0000664 0003717 0003717 00000072775 13627151544 016273 0 ustar gap-jenkins gap-jenkins [aclib] 3 The catalog of almost crystallographic groups
[Up] [Previous] [Next] [Index]
3 The catalog of almost crystallographic groups
Sections
- Rational matrix groups
- Polycyclically presented groups
- More about the type and the defining parameters
- The electronic versus the printed library
This chapter introduces the access functions to the catalog of
3- and 4-dimensional crystallographic groups. This catalog is an
electronic version of the classification obtained in KD.
The following three main functions are available to access the library
of almost crystallographic groups as rational matrix groups.
AlmostCrystallographicGroup( dim, type, parameters )
AlmostCrystallographicDim3( type, parameters )
AlmostCrystallographicDim4( type, parameters )
dim is the dimension of the required group. Thus dim must be
either 3 or 4. The inputs type and parameters are used to define
the desired group as described in KD. We outline the possible
choices for type and parameters here briefly. A more extended
description is given later in Section More about almost crystallographic groups or can be obtained from KD.
type specifies the type of the required group. There are 17 types
in dimension 3 and 95 types in dimension 4. The input type can either
be an integer defining the position of the desired type among all types;
that is, in this case type is a number in [1..17] in dimension 3 or a
number in [1..95] in dimension 4. Alternatively, type can be a string
defining the desired type. In dimension 3 the possible strings are
"01", "02", …, "17". In dimension 4 the possible strings
are listed in the list ACDim4Types and thus can be accessed from GAP.
parameters is a list of integers. Its length depends on the type of
the chosen group. The lists ACDim3Param and ACDim4Param contain
at position i the length of the parameter list for the type number i.
Every list of integers of this length is a valid parameter input.
Alternatively, one can input false instead of a parameter list. Then
GAP will chose a random parameter list of suitable length.
gap> G := AlmostCrystallographicGroup( 4, 50, [ 1, -4, 1, 2 ] );
<matrix group of size infinity with 5 generators>
gap> DimensionOfMatrixGroup( G );
5
gap> FieldOfMatrixGroup( G );
Rationals
gap> GeneratorsOfGroup( G );
[ [ [ 1, 0, -1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, 1/2, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 1 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, -4, 1, 0, 1/2 ], [ 0, 0, -1, 0, 0 ], [ 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 1, 1/4 ], [ 0, 0, 0, 0, 1 ] ] ]
gap> G.1;
[ [ 1, 0, -1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ]
gap> ACDim4Types[50];
"076"
gap> ACDim4Param[50];
4
All the almost crystallographic groups considered in this package are
polycyclic. Hence they have a polycyclic presentation and this can be
used to facilitate efficient computations with the groups. To obtain the
polycyclic presentation of an almost crystallographic group we supply the
following functions. Note that the share package Polycyclic must be
installed to use these functions.
AlmostCrystallographicPcpGroup( dim, type, parameters )
AlmostCrystallographicPcpDim3( type, parameters )
AlmostCrystallographicPcpDim4( type, parameters )
The input is the same as for the corresponding matrix group functions.
The output is a pcp group isomorphic to the corresponding matrix group.
An explicit isomorphism from an almost crystallographic matrix group
to the corresponding pcp group can be obtained by the following function.
IsomorphismPcpGroup( G )
We can use the polycyclic presentations of almost crystallographic
groups to exhibit structure information on these groups. For example,
we can determine their Fitting subgroup and ask group-theoretic
questions about this nilpotent group. The factor G / Fit(G) of an
almost crystallographic group G is called holonomy group. We
provide access to this factor of a pcp group via the following
functions. Let G be an almost crystallographic pcp group.
HolonomyGroup( G )
NaturalHomomorphismOnHolonomyGroup( G )
The following example shows applications of these functions.
gap> G := AlmostCrystallographicPcpGroup( 4, 50, [ 1, -4, 1, 2 ] );
Pcp-group with orders [ 4, 0, 0, 0, 0 ]
gap> Cgs(G);
[ g1, g2, g3, g4, g5 ]
gap> F := FittingSubgroup( G );
Pcp-group with orders [ 0, 0, 0, 0 ]
gap> Centre(F);
Pcp-group with orders [ 0, 0 ]
gap> LowerCentralSeries(F);
[ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0 ],
Pcp-group with orders [ ] ]
gap> UpperCentralSeries(F);
[ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0, 0 ],
Pcp-group with orders [ ] ]
gap> MinimalGeneratingSet(F);
[ g2, g3, g4 ]
gap> H := HolonomyGroup( G );
Pcp-group with orders [ 4 ]
gap> hom := NaturalHomomorphismOnHolonomyGroup( G );
[ g1, g2, g3, g4, g5 ] -> [ g1, identity, identity, identity, identity ]
gap> U := Subgroup( H, [Pcp(H)[1]^2] );
Pcp-group with orders [ 2 ]
gap> PreImage( hom, U );
Pcp-group with orders [ 2, 0, 0, 0, 0 ]
Each group from this library knows that it is almost crystallographic and,
additionally, it knows its type and defining parameters.
AlmostCrystallographicInfo( G )
This attribute is set for groups from the library only. It is not possible
at current to determine the type and the defining parameters for an arbitrary
almost crystallographic groups which is not defined by the library access
functions.
gap> G := AlmostCrystallographicGroup( 4, 70, false );
<matrix group of size infinity with 5 generators>
gap> IsAlmostCrystallographic(G);
true
gap> AlmostCrystallographicInfo(G);
rec( dim := 4, type := 70, param := [ 1, -4, 1, 2, -3 ] )
gap> G := AlmostCrystallographicPcpGroup( 4, 70, false );
Pcp-group with orders [ 6, 0, 0, 0, 0 ]
gap> IsAlmostCrystallographic(G);
true
gap> AlmostCrystallographicInfo(G);
rec( dim := 4, type := 70, param := [ -3, 2, 5, 1, 0 ] )
We consider the types of almost crystallographic groups in more detail. The
almost crystallographic groups in dimensions 3 and 4 fall into three families
- (1)
- 3-dimensional almost crystallographic groups.
- (2)
- 4-dimensional almost crystallographic groups with a
Fitting subgroup of class 2.
- (3)
- 4-dimensional almost crystallographic groups with a
Fitting subgroup of class 3.
These families are split up further into subfamilies in KD and to
each subfamily is assigned a type; that is, a string which is used to
identify the subfamily. As mentioned above, for the 3-dimensional almost
crystallographic groups the type is a string representing the numbers from
1 to 17, i.e. the available types are "01", "02", …, "17".
For the 4-dimensional almost crystallographic groups with a Fitting subgroup
of class 2 the type is a string of 3 or 4 characters. In general, a string of
3 characters representing
the number of the table entry in KD is used. So possible types are
"001", "002", …. The reader is warned however that not all
possible numbers are used, e.g. there are no groups of type "016". Also,
the types do not appear in their natural order in KD. Moreover, for
certain numbers there is more than one family of groups listed in KD.
For example, the 3 families of groups corresponding to number 19 on pages
179-180 of KD have types "019", "019b" and "019c" (the order is
the one given in KD).
For the last category of groups, the 4-dimensional almost crystallographic
groups with a Fitting subgroup of class 3, the type is a string of 2 or 3
characters, where the first character is always the letter "B". This "B"
is followed by the number of the table entry as found in KD,
eventually followed by a "b" or "c" as in the previous case.
For each type of almost crystallographic group contained in the library
there exists a function taking a parameter list as input and returning
the desired matrix or pcp group. These functions can be accessed
from GAP using the lists ACDim3Funcs, ACDim4Funcs, ACPcpDim3Funcs
and ACPcpDim4Funcs which consist of the corresponding functions.
Although we include these direct access functions here for completeness,
we note that the user should in general use the higher-level functions
introduced above to obtain almost crystallographic groups from the
library. In particular, these low-level access functions return matrix
or pcp groups, but the almost crystallographic info flags will not be
attached to them.
gap> ACDim3Funcs[15];
function( k1, k2, k3, k4 ) ... end
gap> ACDim3Funcs[15](1,1,1,1);
<matrix group with 5 generators>
gap> ACPcpDim3Funcs[1](1);
Pcp-group with orders [ 0, 0, 0 ]
The package aclib can be considered as the electronic version of
Chapter 7 of KD. In this section we outline the relationship
between the library presented in this manual and the printed version
in KD. First we consider an example. At page 175 of KD,
we find the following groups in the table starting with entry ``13''.
13. Q=P2/c
| λ(α)= | ⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝ | | | ⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠ | λ(β)= | ⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝ | | | ⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠ | |
|
| A={(k1,…,k6)|| k1=0, k2,…, k5 ∈ 2Z, k6 ∈ Z} |
|
AB-groups:
∀k > 0, k ≡ 0 mod 2, (k,0,1,0,1,0)
The number ``13'' at the beginning of this entry is the type of the
almost crystallographic group in this library. This family of groups
with type 13 depends on 6 parameters k1, k2, …, k6 and these
are the parameters list in this library. The rational matrix
representation in GAP corresponds exactly to the printed version in
KD where it is named λ. In the example below, we consider
the group with parameters (k1,k2,k3,k4,k5,k6)=(8,0,1,0,1,0).
gap> G:=AlmostCrystallographicDim4("013",[8,0,1,0,1,0]);
<matrix group with 6 generators>
gap> G.5;
[ [ 1, 4, 0, 0, 1/2 ], [ 0, -1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, -1, 1/2 ], [ 0, 0, 0, 0, 1 ] ]
gap> G.6;
[ [ 1, 8, 0, 0, 1/2 ], [ 0, -1, 0, 0, 0 ], [ 0, 0, -1, 0, 0 ],
[ 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 1 ] ]
For a 4-dimensional almost crystallographic group the matrix group is
built up such that { a, b, c, d, α, β, γ} as described
in KD forms the defining generating set of G. For certain types
the elements α, β or γ may not be present.
Similarly, for a 3-dimensional group we have the generating set { a, b, c, α, β} and α and β may be absent.
To obtain a polycyclic generating sequence from the defining generators
of the matrix group we have to order the elements in the generating set
suitably. For this purpose we take the subsequence of (γ, β, α, a, b, c, d) of those generators which are present in the
defining generating set of the matrix group. This new ordering of the
generators is then used to define a polycyclic presentation of the given
almost crystallographic group.
[Up] [Previous] [Next] [Index]
aclib manual
January 2020
aclib-1.3.2/htm/CHAP002.htm 0000664 0003717 0003717 00000013762 13627151544 016261 0 ustar gap-jenkins gap-jenkins [aclib] 2 Algorithms for almost crystallographic groups
[Up] [Previous] [Next] [Index]
2 Algorithms for almost crystallographic groups
Sections
- Properties of almost crystallographic groups
- Betti numbers
- Determination of certain extensions
This chapter presents a variety of algorithms for almost crystallographic
groups. In most cases, they assume a polycyclically presented group as
input; in particular, the input groups must be polycyclic in this case.
The methods described here supplement the methods of the Polycyclic
package for polycyclically presented groups. Many of the functions in this
chapter are based on methods of the Polycyclic package and thus this
package must be installed to use the functions introduced here. We refer to
the Polycyclic package for further information on polycyclic
presentations.
IsAlmostCrystallographic( G ) P
This function checks if a polycyclically presented group G is almost
crystallographic; that is, it checks if G is nilpotent-by-finite and
has no non-trivial finite normal subgroup.
IsAlmostBieberbachGroup( G ) P
This function checks if a polycyclically presented group G is almost
Bieberbach; that is, it checks if G is nilpotent-by-finite and torsion
free.
Let G be a polycyclically presented and torsion free group of Hirsch
length n. Then we can compute the Betti numbers βi(G) for i ∈ {0, 1, 2, n−2, n−1, n}. If n ≤ 6, then we can compute all Betti
numbers βi(G) for 0 ≤ i ≤ 6 of G. We introduce the following
functions for this purpose and we refer to BRO for the details on
the orientation module and the Betti numbers.
OrientationModule( G ) F
This function determines the orientation module of the polycyclically
presented group G; that is, it returns a list of matrices m1, …, mn ≤ GL( 1, Z) which are the images of the 'Igs(G)' in their action
on the orientation module.
BettiNumber( G, m ) F
This function returns the mth Betti number of the polycyclically presented
torsion free group G if m ∈ {0, 1, 2, n−2, n−1, n}, where n is the
Hirsch length of G.
BettiNumbers( G ) A
This function returns the Betti number of the polycyclically presented
torsion free group G if the Hirsch length of G is smaller than 7.
Let G be a polycyclically presented almost crystallographic group. We want
to check the existence of certain extensions of G.
First, it is well-known that the equivalence classes of extensions of G
correspond to the second cohomology group of G. This cohomology group can
be computed using the methods of the Polycyclic package for any
explicitly given module of G. Further, we can construct a polycyclic
presentation for each cocycle of the second cohomology group. We give an
example for such a computation below.
However, we may be interested in certain extensions only; for example,
the torsion free extensions are often of particular interest. If the
second cohomology group is finite, then we can compute a polycyclic
presentation for each element of this group and check the resulting group
for torsion freeness. But if the second cohomology group is infinite, then
this approach is not available. Hence we introduce the following special
method to cover this and related applications.
HasExtensionOfType( G, torsionfree, minimalcentre ) F
Suppose that G is a polycyclically presented almost crystallographic group
with Fitting subgroup N. This function checks if there is a G-module
M ≅ Z which is centralized by N such that there exists a torsion
free extension of M by G (if the flag torsionfree is true) or an
extension E with Z(Fitt(E)) = M (if the flag minimalcentre is true)
or an extension which satisfies both conditions (if both flags are true).
We note that the existence of such extensions is of interest in the
determination of extensions which are almost Bieberbach groups. We refer
to DE1 for a more detailed account of this application and for
further results of a similar nature.
[Up] [Previous] [Next] [Index]
aclib manual
January 2020
aclib-1.3.2/gap/pcpgrp3.gi 0000664 0003717 0003717 00000024726 13627151544 016472 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W pcpgrp3.gi Karel Dekimpe
#W Bettina Eick
##
## This file contains the 3-dimensional almost crystallographic groups
## as pcp groups. There are 17 types groups.
##
ACPcpGroupDim3Nr01 := function (k1)
local FTL;
FTL := FromTheLeftCollector( 3 );
SetConjugate( FTL, 2, 1, [2,1, 3,k1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr02 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 4 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,k4] );
SetConjugate( FTL, 2, 1, [2,-1, 3,0, 4,k2] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,k3] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1] );
SetConjugate( FTL, 3, 2, [3,1, 4,k1] );
SetConjugate( FTL, 4, 2, [3,0, 4,1] );
SetConjugate( FTL, 4, 3, [4,1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr03 := function (k1, k2)
local FTL;
FTL := FromTheLeftCollector( 4 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,-k2] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,0] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1] );
SetConjugate( FTL, 3, 2, [3,1, 4,k1] );
SetConjugate( FTL, 4, 2, [3,0, 4,1] );
SetConjugate( FTL, 4, 3, [4,1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr04 := function (k1, k2)
local FTL;
FTL := FromTheLeftCollector( 4 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,1, 3,0, 4,k2] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,2*k2] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,-k1] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1] );
SetConjugate( FTL, 3, 2, [3,1, 4,2*k1] );
SetConjugate( FTL, 4, 2, [3,0, 4,1] );
SetConjugate( FTL, 4, 3, [4,1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr05 := function (k1, k2)
local FTL;
FTL := FromTheLeftCollector( 4 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0] );
SetConjugate( FTL, 2, 1, [2,0, 3,1, 4,-k2] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,-k2] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1] );
SetConjugate( FTL, 3, 2, [3,1, 4,k1] );
SetConjugate( FTL, 4, 2, [3,0, 4,1] );
SetConjugate( FTL, 4, 3, [4,1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr06 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,-k4] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,-k2] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,k4] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,k2] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,k3] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr07 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,-1, 5,-2*k3 - k4] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,k1 - k2] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,k3] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,k2] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,-2*(k3 + k4)] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1] );
SetConjugate( FTL, 4, 3, [4,1, 5,2*k1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr08 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,-k3] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,-1, 5,-2*k1 + k2 - 2*k3 - k4] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,-2*k3] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,-k1] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,k4] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,k1 + 2*k3] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,-k1 + 2*k2 - 2*k3] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1] );
SetConjugate( FTL, 4, 3, [4,1, 5,2*k1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr09 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,-k4] );
SetConjugate( FTL, 3, 1, [3,0, 4,1, 5,-k3] );
SetConjugate( FTL, 4, 1, [3,1, 4,0, 5,-k3] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,k4] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,k2] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,-k2 + 2*k3] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr10 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 4 );
SetRelativeOrder( FTL, 1, 4 );
SetPower( FTL, 1, [2,0, 3,0, 4,k4] );
SetConjugate( FTL, 2, 1, [2,0, 3,-1, 4,k3] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,-k2] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1] );
SetConjugate( FTL, 3, 2, [3,1, 4,k1] );
SetConjugate( FTL, 4, 2, [3,0, 4,1] );
SetConjugate( FTL, 4, 3, [4,1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr11 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0] );
SetConjugate( FTL, 2, 1, [2, 3 , 3,0, 4,0, 5,-k4] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,-k2 - k3] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,k4] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,k3] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,-k2] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr12 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,k3] );
SetConjugate( FTL, 2, 1, [2, 3 , 3,0, 4,-1, 5,-k2 - k3 - k4] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,2*k3] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,-k1] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,k4] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,k1 - k2 - 2*k3] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,-k2] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1] );
SetConjugate( FTL, 4, 3, [4,1, 5,2*k1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr13 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 4 );
SetRelativeOrder( FTL, 1, 3 );
SetPower( FTL, 1, [2,0, 3,0, 4,k4] );
SetConjugate( FTL, 2, 1, [2,-1, 3,-1, 4,k2 + k3] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,-k2] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1] );
SetConjugate( FTL, 3, 2, [3,1, 4,k1] );
SetConjugate( FTL, 4, 2, [3,0, 4,1] );
SetConjugate( FTL, 4, 3, [4,1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr14 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0] );
SetConjugate( FTL, 2, 1, [2, 2 , 3,0, 4,0, 5,-k4] );
SetConjugate( FTL, 3, 1, [3,0, 4,-1, 5,-k2] );
SetConjugate( FTL, 4, 1, [3,-1, 4,0, 5,k2] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1] );
SetRelativeOrder( FTL, 2, 3 );
SetPower( FTL, 2, [3,0, 4,0, 5,k4] );
SetConjugate( FTL, 3, 2, [3,-1, 4,-1, 5,k2 + k3] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,-k2] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr15 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0] );
SetConjugate( FTL, 2, 1, [2, 2 , 3,0, 4,0, 5,-k4] );
SetConjugate( FTL, 3, 1, [3,0, 4,1, 5,-k2] );
SetConjugate( FTL, 4, 1, [3,1, 4,0, 5,-k2] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1] );
SetRelativeOrder( FTL, 2, 3 );
SetPower( FTL, 2, [3,0, 4,0, 5,k4] );
SetConjugate( FTL, 3, 2, [3,-1, 4,-1, 5,k1 + 3*k2 - k3] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,-k1 - 3*k2 + 2*k3] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr16 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 4 );
SetRelativeOrder( FTL, 1, 6 );
SetPower( FTL, 1, [2,0, 3,0, 4,k4] );
SetConjugate( FTL, 2, 1, [2,0, 3,-1, 4,k3] );
SetConjugate( FTL, 3, 1, [2,1, 3,1, 4,k1 - k2 - k3] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1] );
SetConjugate( FTL, 3, 2, [3,1, 4,k1] );
SetConjugate( FTL, 4, 2, [3,0, 4,1] );
SetConjugate( FTL, 4, 3, [4,1] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim3Nr17 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0] );
SetConjugate( FTL, 2, 1, [2, 5 , 3,0, 4,0, 5,-k4] );
SetConjugate( FTL, 3, 1, [3,0, 4,1, 5,-k3] );
SetConjugate( FTL, 4, 1, [3,1, 4,0, 5,-k3] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1] );
SetRelativeOrder( FTL, 2, 6 );
SetPower( FTL, 2, [3,0, 4,0, 5,k4] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,k3] );
SetConjugate( FTL, 4, 2, [3,1, 4,1, 5,k1 - k2 - k3] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1] );
return PcpGroupByCollector(FTL);
end;
#############################################################################
##
## some small helpers
##
ACPcpDim3Funcs := [ ACPcpGroupDim3Nr01, ACPcpGroupDim3Nr02,
ACPcpGroupDim3Nr03,
ACPcpGroupDim3Nr04, ACPcpGroupDim3Nr05, ACPcpGroupDim3Nr06,
ACPcpGroupDim3Nr07, ACPcpGroupDim3Nr08, ACPcpGroupDim3Nr09,
ACPcpGroupDim3Nr10, ACPcpGroupDim3Nr11, ACPcpGroupDim3Nr12,
ACPcpGroupDim3Nr13, ACPcpGroupDim3Nr14, ACPcpGroupDim3Nr15,
ACPcpGroupDim3Nr16, ACPcpGroupDim3Nr17 ];
MakeReadOnlyGlobal( "ACPcpDim3Funcs" );
aclib-1.3.2/gap/matgrp.gi 0000664 0003717 0003717 00000010657 13627151544 016404 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W matgrp.gi Bettina Eick
##
## This file contains the header functions to handle the 3- and
## 4-dimensional almost crystallographic integral matrix groups.
##
#############################################################################
##
#F AlmostCrystallographicDim3( type, param )
##
InstallGlobalFunction( AlmostCrystallographicDim3,
function( type, param )
local g, G, info;
# type is integer or string
if IsString( type ) then
g := Int( type );
elif IsInt( type ) then
g := type;
else
g := false;
fi;
if not g in [1..17] then
Error("type does not define a valid ac-group type");
fi;
# check parameters
if IsBool( param ) then
param := List( [1..ACDim3Param[g]], x -> Random(Integers) );
if param[1] = 0 then param[1] := Random(Integers)^2 + 1; fi;
elif IsInt( param ) then
param := List( [1..ACDim3Param[g]], x -> param );
elif Length( param ) <> ACDim3Param[g] then
Error("parameter should be a list of length ", ACDim3Param[g] );
fi;
# get group
if Length( param ) = 4 then
G := ACDim3Funcs[g](param[1], param[2], param[3], param[4]);
elif Length(param) = 2 then
G := ACDim3Funcs[g](param[1], param[2] );
elif Length(param) = 1 then
G := ACDim3Funcs[g](param[1]);
fi;
# set some information
info := rec( dim := 3, type := g, param := param );
SetAlmostCrystallographicInfo( G, info );
SetIsAlmostCrystallographic( G, true );
SetDimensionOfMatrixGroup( G, 4 );
SetFieldOfMatrixGroup( G, Rationals );
SetSize( G, infinity );
return G;
end );
#############################################################################
##
#F AlmostCrystallographicDim4( type, param )
##
InstallGlobalFunction( AlmostCrystallographicDim4,
function( type, param )
local g, G, info;
# type is integer or string
if IsString( type ) then
g := Position( ACDim4Types, type );
elif IsInt( type ) then
g := type;
else
g := false;
fi;
if not g in [1..95] then
Error("type does not define a valid ac-group type");
fi;
# check parameters
if IsBool( param ) then
param := List( [1..ACDim4Param[g]], x -> Random(Integers) );
if param[1] = 0 then param[1] := Random(Integers)^2 + 1; fi;
elif IsInt( param ) then
param := List( [1..ACDim4Param[g]], x -> param );
elif Length( param ) <> ACDim4Param[g] then
Error("parameter should be a list of length ", ACDim4Param[g] );
fi;
# get group
if Length(param) = 7 then
G := ACDim4Funcs[g]( param[1], param[2], param[3], param[4],
param[5], param[6], param[7]);
elif Length(param) = 6 then
G := ACDim4Funcs[g]( param[1], param[2], param[3], param[4],
param[5], param[6]);
elif Length(param) = 5 then
G := ACDim4Funcs[g]( param[1], param[2], param[3], param[4],
param[5]);
elif Length(param) = 4 then
G := ACDim4Funcs[g]( param[1], param[2], param[3], param[4]);
elif Length(param) = 3 then
G := ACDim4Funcs[g]( param[1], param[2], param[3]);
fi;
# set some information
info := rec( dim := 4, type := g, param := param );
SetAlmostCrystallographicInfo( G, info );
SetIsAlmostCrystallographic( G, true );
SetDimensionOfMatrixGroup( G, 5 );
SetFieldOfMatrixGroup( G, Rationals );
SetSize( G, infinity );
return G;
end );
#############################################################################
##
#F AlmostCrystallographicGroup( dim, type, param )
##
InstallGlobalFunction( AlmostCrystallographicGroup,
function( dim, type, param )
if dim = 3 then
return AlmostCrystallographicDim3( type, param );
elif dim = 4 then
return AlmostCrystallographicDim4( type, param );
else
Error("dimension must be 3 or 4");
fi;
end );
#############################################################################
##
#F IsAlmostCrystallographic( )
##
InstallMethod( IsAlmostCrystallographic, "for groups", true,
[IsGroup ], 0,
function( G )
if HasAlmostCrystallographicInfo( G ) then
return true;
else
Print("sorry - cannot check this property \n");
return fail;
fi;
end );
aclib-1.3.2/gap/pcpgrp.gi 0000664 0003717 0003717 00000015620 13627151544 016400 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W pcpgrp.gi Bettina Eick
##
## This file contains the header functions to handle the 3- and
## 4-dimensional almost crystallographic pcp groups.
##
#############################################################################
##
#F IsAlmostCrystallographic( )
##
InstallMethod( IsAlmostCrystallographic, "for pcp groups", true,
[IsPcpGroup ], 0,
function( G )
if HasAlmostCrystallographicInfo( G ) then return true; fi;
return (IsInt(Index(G,FittingSubgroup(G))) and
Size(NormalTorsionSubgroup(G)) = 1);
end );
#############################################################################
##
#F IsAlmostBieberbachGroup( )
##
InstallMethod( IsAlmostBieberbachGroup, "for pcp groups", true,
[IsPcpGroup ], 0,
function( G )
return IsInt(Index(G,FittingSubgroup(G))) and IsTorsionFree(G);
end );
#############################################################################
##
#F IsolatorSubgroup( G, N )
##
InstallGlobalFunction( IsolatorSubgroup, function( G, N )
local nat, F, T;
if not IsNormal( G, N ) then return fail; fi;
nat := NaturalHomomorphismByNormalSubgroup( G, N );
F := Image( nat );
T := TorsionSubgroup( F );
if IsBool( T ) then return fail; fi;
return PreImage( nat, T );
end);
#############################################################################
##
#F AlmostCrystallographicPcpDim3( type, param )
##
InstallGlobalFunction( AlmostCrystallographicPcpDim3,
function( type, param )
local g, G, info;
# type is integer or string
if IsString( type ) then
g := Int( type );
elif IsInt( type ) then
g := type;
else
g := false;
fi;
if not g in [1..17] then
Error("type does not define a valid ac-group type");
fi;
# check parameters
if IsBool( param ) then
param := List( [1..ACDim3Param[g]], x -> Random(Integers) );
if param[1] = 0 then param[1] := Random(Integers)^2+1; fi;
elif IsInt( param ) then
param := List( [1..ACDim3Param[g]], x -> param );
elif Length( param ) <> ACDim3Param[g] then
Error("parameter should be a list of length ", ACDim3Param[g] );
fi;
# get group
if Length( param ) = 4 then
G := ACPcpDim3Funcs[g](param[1], param[2], param[3], param[4]);
elif Length(param) = 2 then
G := ACPcpDim3Funcs[g](param[1], param[2] );
elif Length(param) = 1 then
G := ACPcpDim3Funcs[g](param[1]);
fi;
# set some information
info := rec( dim := 3, type := g, param := param );
SetAlmostCrystallographicInfo( G, info );
SetIsAlmostCrystallographic( G, true );
SetSize( G, infinity );
return G;
end );
#############################################################################
##
#F AlmostCrystallographicPcpDim4( type, param )
##
InstallGlobalFunction( AlmostCrystallographicPcpDim4,
function( type, param )
local g, G, info;
# type is integer or string
if IsString( type ) then
g := Position( ACDim4Types, type );
elif IsInt( type ) then
g := type;
else
g := false;
fi;
if not g in [1..95] then
Error("type does not define a valid ac-group type");
fi;
# check parameters
if IsBool( param ) then
param := List( [1..ACDim4Param[g]], x -> Random(Integers) );
if param[1] = 0 then param[1] := Random(Integers)^2+1; fi;
elif IsInt( param ) then
param := List( [1..ACDim4Param[g]], x -> param );
elif Length( param ) <> ACDim4Param[g] then
Error("parameter should be a list of length ", ACDim4Param[g] );
fi;
# get group
if Length(param) = 7 then
G := ACPcpDim4Funcs[g]( param[1], param[2], param[3], param[4],
param[5], param[6], param[7]);
elif Length(param) = 6 then
G := ACPcpDim4Funcs[g]( param[1], param[2], param[3], param[4],
param[5], param[6]);
elif Length(param) = 5 then
G := ACPcpDim4Funcs[g]( param[1], param[2], param[3], param[4],
param[5]);
elif Length(param) = 4 then
G := ACPcpDim4Funcs[g]( param[1], param[2], param[3], param[4]);
elif Length(param) = 3 then
G := ACPcpDim4Funcs[g]( param[1], param[2], param[3]);
fi;
# set some information
info := rec( dim := 4, type := g, param := param );
SetAlmostCrystallographicInfo( G, info );
SetIsAlmostCrystallographic( G, true );
SetSize( G, infinity );
return G;
end );
#############################################################################
##
#F AlmostCrystallographicPcpGroup( dim, type, param )
##
InstallGlobalFunction( AlmostCrystallographicPcpGroup,
function( dim, type, param )
if dim = 3 then
return AlmostCrystallographicPcpDim3( type, param );
elif dim = 4 then
return AlmostCrystallographicPcpDim4( type, param );
else
Error("dimension must be 3 or 4");
fi;
end );
#############################################################################
##
#A FittingSubgroup( < G > )
##
InstallMethod( FittingSubgroup, "for ac pcp groups", true,
[IsPcpGroup and HasAlmostCrystallographicInfo], 0,
function( G )
local pcp, rel, sub, F;
pcp := Pcp( G );
rel := RelativeOrdersOfPcp( pcp );
sub := Filtered( [1..Length(pcp)], x -> rel[x] = 0 );
F := Subgroup( G, pcp{sub} );
SetIsNilpotentGroup( F, true );
return F;
end );
#############################################################################
##
#F NaturalHomomorphismOnHolonomyGroup( < G > )
#F HolonomyGroup( < G > )
##
InstallMethod( NaturalHomomorphismOnHolonomyGroup, "for ac pcp groups", true,
[IsPcpGroup and IsAlmostCrystallographic], 0,
function( G )
return NaturalHomomorphismByNormalSubgroup(G,FittingSubgroup(G));
end );
InstallMethod( HolonomyGroup, "for ac pcp groups", true,
[IsPcpGroup and IsAlmostCrystallographic], 0,
function( G )
return Image( NaturalHomomorphismOnHolonomyGroup(G) );
end );
#############################################################################
##
#F IsomorphismPcpGroup( )
##
InstallMethod( IsomorphismPcpGroup, "for ac groups", true,
[IsMatrixGroup and HasAlmostCrystallographicInfo], 0,
function( G )
local info, H, gensH, gensG, l, d, newsG, hom;
# get the corresponding pcp group
info := AlmostCrystallographicInfo( G );
H := AlmostCrystallographicPcpGroup( info.dim, info.type, info.param );
gensH := GeneratorsOfGroup(H);
gensG := GeneratorsOfGroup(G);
# sort the generators of G according to the generators of H
l := Length( gensG );
d := info.dim;
newsG := Concatenation( Reversed( gensG{[d+1..l]} ), gensG{[1..d]} );
hom := GroupHomomorphismByImagesNC( G, H, newsG, gensH );
SetIsInjective( hom, true );
SetIsSurjective( hom, true );
return hom;
end );
aclib-1.3.2/gap/extend.gi 0000664 0003717 0003717 00000023310 13627151544 016367 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W extend.gi Karel Dekimpe
#W Bettina Eick
##
#############################################################################
##
#F FiniteSubgroupsAreCyclic( G )
##
FiniteSubgroupsAreCyclic := function( G )
local fin, rep, flg;
fin := FiniteSubgroupClasses( G );
rep := List( fin, Representative );
flg := ForAll( rep, IsCyclic );
return flg;
end;
#############################################################################
##
#F AllActionsHolonomy( G )
##
## Computes all actions of G on Z which can lead to a torsion-free
## extension centralizing Fit(G). Thus all finite subgroups of G and Fit(G)
## centralizes Z and the kernel of the action has index at most 2.
##
AllActionsHolonomy := function( G )
local mats, acts, fins, reps, gens, U, hom, F, low, H, i;
# first we add the trivial action
mats := List( Igs(G), x -> IdentityMat(1));
acts := [mats];
# determine the subgroup known to centralize Z
U := FittingSubgroup(G);
# determine all subgroups of index 2 in G/U
hom := NaturalHomomorphismByNormalSubgroup( G, U );
F := Image( hom );
low := LowIndexSubgroupClasses( F, 2 );
low := List( low, Representative );
low := List( low, x -> PreImage( hom, x ) );
# for each subgroup we get one action
for H in low do
mats := List( Igs(G), x -> IdentityMat(1));
for i in [1..Length(Igs(G))] do
if not Igs(G)[i] in H then
mats[i] := - mats[i];
fi;
od;
Add( acts, mats );
od;
return acts;
end;
#############################################################################
##
#F AllActionsForTorsionFreeExtension( G )
##
## Computes all actions of G on Z which can lead to a torsion-free
## extension centralizing Fit(G). Thus all finite subgroups of G and Fit(G)
## centralizes Z and the kernel of the action has index at most 2.
##
AllActionsForTorsionFreeExtension := function( G )
local mats, acts, fins, reps, gens, U, hom, F, low, H, i;
# first we add the trivial action
mats := List( Igs(G), x -> IdentityMat(1));
acts := [mats];
# determine the subgroup known to centralize Z
fins := FiniteSubgroupClasses( G );
reps := List( fins, Representative );
gens := Flat( List( reps, Igs ) );
Append( gens, GeneratorsOfGroup( FittingSubgroup(G) ) );
U := NormalClosure( G, Subgroup( G, gens ) );
if Index( G, U ) = 1 then return acts; fi;
# determine all subgroups of index 2 in G/U
hom := NaturalHomomorphismByNormalSubgroup( G, U );
F := Image( hom );
low := LowIndexSubgroupClasses( F, 2 );
low := List( low, Representative );
low := List( low, x -> PreImage( hom, x ) );
# for each subgroup we get one action
for H in low do
mats := List( Igs(G), x -> IdentityMat(1));
for i in [1..Length(Igs(G))] do
if not Igs(G)[i] in H then
mats[i] := - mats[i];
fi;
od;
Add( acts, mats );
od;
return acts;
end;
#############################################################################
##
#F ExpandedTail( tail, d )
##
ExpandedTail := function( t, d )
local r, i;
r := List( [1..d], x -> 0 );
for i in [1..Length(t)] do
if IsBound( t[i] ) then r[i] := t[i][1][1]; fi;
od;
return r;
end;
#############################################################################
##
#F HasTorsionFreeExtension( G, CR )
##
## Check if G has a torsion-free extension. We assume that all finite
## subgroups of G are cyclic (otherwise, no torsion-free extension exists).
## CR ist a record describing the second cohomology group of G by Z.
##
HasTorsionFreeExtension := function( G, CR )
local fins, reps, repp, cc, d, subs, U, g, e, p, w, v, i, t, r, pr,
f, V, spcs, sub;
# compute the finite subgroups of prime order up to conjugacy
fins := FiniteSubgroupClasses( G );
reps := List( fins, Representative );
repp := Filtered( reps, x -> IsPrime( Size(x) ) );
if Length( repp ) = 0 then return true; fi;
# extract second cohomology
cc := CR.twocohom;
# for each finite subgroup of prime order compute the corresponding
# relation on cc.gcc
subs := [];
for U in repp do
g := GeneratorsOfGroup( U )[1];
e := Exponents( g );
p := Order( g );
w := rec( word := e, tail := [] );
v := ShallowCopy( w );
for i in [2..p] do
v := CollectedTwoCR( CR, v, w );
od;
r := ExpandedTail( v.tail, Length( CR. enumrels ) );
sub := NullspaceIntMod( cc.factor.prei, r, p );
Add( subs, sub );
od;
e := Lcm( List( repp, Size ) );
d := Length( cc.factor.prei );
return SizeOfUnionMod( subs, e ) < e^d;
end;
#############################################################################
##
#F HasMinimalCentreExtension( G, CR )
##
## Check if there is an extension of G that has minimal centre; that is,
## Z(Fit(E)) = Z. This is only implemented for groups G with abelian
## Fitting subgroup.
##
HasMinimalCentreExtension := function( G, CR )
local cc, d, t, F, f, g, l, c, i, j, v, w, a, b;
# get cohomology and generic elements
cc := CR.twocohom;
d := Length( CR.enumrels );
t := List( [1..Length(cc.gcc)], x -> Indeterminate( Integers, x ) );
t := t * cc.gcc;
# get Fitting subgroup generators
F := FittingSubgroup( G );
# the abelian case is more effective - consider this first
if IsAbelian( F ) then
f := GeneratorsOfGroup( F );
l := Length( f );
# find tails of commutators
c := NullMat( l, l );
for i in [1..l] do
for j in [i+1..l] do
v := rec( word := Exponents( f[i]^-1 ), tail := [] );
w := rec( word := Exponents( f[j]^-1 ), tail := [] );
v := CollectedTwoCR( CR, v, w );
w := rec( word := Exponents( f[i] ), tail := [] );
v := CollectedTwoCR( CR, v, w );
w := rec( word := Exponents( f[j] ), tail := [] );
v := CollectedTwoCR( CR, v, w );
b := ExpandedTail( v.tail, d );
a := b * t;
c[i][j] := a;
c[j][i] := -a;
od;
od;
# get determinant
c := GenericDeterminantMat( c );
return Length( ExtRepPolynomialRatFun(c) ) <> 0;
fi;
# the non-abelian case
f := MinimalGeneratingSet( F );
g := GeneratorsOfGroup( Centre( F ) );
if Length(f) < Length( g ) then return false; fi;
# find tails of commutators
c := NullMat( Length(f), Length(g) );
for i in [1..Length(f)] do
for j in [1..Length(g)] do
v := rec( word := Exponents( f[i]^-1 ), tail := [] );
w := rec( word := Exponents( g[j]^-1 ), tail := [] );
v := CollectedTwoCR( CR, v, w );
w := rec( word := Exponents( f[i] ), tail := [] );
v := CollectedTwoCR( CR, v, w );
w := rec( word := Exponents( g[j] ), tail := [] );
v := CollectedTwoCR( CR, v, w );
b := ExpandedTail( v.tail, d );
c[i][j] := b * t;
od;
od;
# get determinant of rxr submats
Error("not yet implemented");
c := Determinant( c );
c := ExtRepPolynomialRatFun( c );
c := Filtered( c, x -> IsInt(x) );
return ForAny( c, x -> x <> 0 );
end;
#############################################################################
##
#F HasExtensionOfType( G, torfree, mincent )
##
## Let G be a pcp group. This function checks if G has a torsion-free
## extension or an extension with Z(Fit(H)) = Z$ with the free abelian
## module Z.
##
InstallGlobalFunction( HasExtensionOfType, function( G, torfree, mincent )
local mats, CR, found, i;
# if both flags are false, then there is nothing to do
if not torfree and not mincent then return true; fi;
# first check if G is torsion-free or has non-cyclic finite subgrps
Print(" check that all finite subgroups are cyclic\n");
if not FiniteSubgroupsAreCyclic( G ) then return false; fi;
# if G is torsion-free, then all extensions of G are torsion-free
if Length( FiniteSubgroupClasses(G) ) = 1 then torfree := false; fi;
if not torfree and not mincent then return true; fi;
# now loop over actions - the trivial action first
Print(" consider trivial action \n");
mats := List( Igs(G), x -> IdentityMat( 1 ) );
CR := CRRecordByMats( G, mats );
CR.twocohom := TwoCohomologyCR( CR );
if mincent then
found := HasMinimalCentreExtension( G, CR );
#if found <> HasMinimalCentreExtensionByCRRec2( G, CR ) then
# Error("something wrong with min cent");
#fi;
fi;
if (not mincent) or (found and torfree) then
found := HasTorsionFreeExtension( G, CR );
fi;
if found then return true; fi;
# now consider the remaining actions
mats := AllActionsForTorsionFreeExtension( G );
for i in [2..Length(mats)] do
Print(" consider non-trivial action number ",i-1,"\n");
CR := CRRecordByMats( G, mats[i] );
CR.twocohom := TwoCohomologyCR( CR );
if mincent then
found := HasMinimalCentreExtension( G, CR );
#if found <> HasMinimalCentreExtensionByCRRec2( G, CR ) then
# Error("something wrong with min cent");
#fi;
fi;
if (not mincent) or (found and torfree) then
found := HasTorsionFreeExtension( G, CR );
fi;
if found then return true; fi;
od;
# we have not found a suitable extension
return false;
end );
aclib-1.3.2/gap/matgrp4.gi 0000664 0003717 0003717 00000225310 13627151544 016462 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W matgrp4.gi Karel Dekimpe
#W Bettina Eick
##
## This file contains the 4-dimensional almost crystallographic groups
## as integral matrix groups. There are 95 types of groups.
##
ACDim4Nr001 := function ( k1, k2, k3)
local a, b, c, d;
a :=[[1, 0, -k1/2, -k2/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, -k3/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
c :=[[1, k2/2, k3/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
return Group( [a, b, c, d] , IdentityMat(5) );
end;
ACDim4Nr002 := function ( k1, k2, k3, k4, k5, k6, k7)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, -k2/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, -k3/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
c :=[[1, k2/2, k3/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k4, k5, k6, k7/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr003 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, 0, k3, k4/2], [0, -1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr004 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, 0, - k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, 0, k3, k4/2], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr004b := function ( k1, k2, k3)
local a, b, c, d, alfa;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, -k2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, k2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1/2, 2*k3, k2/2, 0], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr005 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k2, k3, k4/2], [0, 0, -1, 0, 0], [0, -1, 0, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr006 := function ( k1, k2, k3)
local a, b, c, d, alfa;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, 0, k2, 0, k3/2], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr007 := function ( k1, k2, k3)
local a, b, c, d, alfa;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1/2, k2, 0, k3/2], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr007b := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, -k2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, k2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k3, -k2/2, 2*k4, 0], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr008 := function ( k1, k2, k3)
local a, b, c, d, alfa;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k2, 0, k3/2], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr009 := function ( k1, k2, k3)
local a, b, c, d, alfa;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1/4 + k2, (3*k1)/4 - k2, 0, k3/2], [0, 0, 1, 0, 0],
[0, 1, 0, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr009b := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, -k2/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, k2/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
c :=[[1, k2/2, -k2/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, k2/4 - k3, (3*k2)/4 - k3, 2*k4, 0], [0, 0, 1, 0, 0],
[0, 1, 0, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr010 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, 0, k3, k4/2], [0, -1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
beta :=[[1, k2, k5, k3, k6/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr011 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, 0, k3, k4/2], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
beta :=[[1, k2, -2*k6, k3, k5/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr012 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k2, k3, k4/2], [0, 0, -1, 0, 0], [0, -1, 0, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
beta :=[[1, k5, 2*k2 - k5, k3, k6/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr013 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1/2 + k2, 0, -2*k6, k3/2 + k6/2], [0, -1, 0, 0, 0],
[0, 0, 1, 0, 0], [0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, k1 + k2, k4, -2*k6, k5/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr014 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a:=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b:=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c:=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d:=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:=[[1, k1/2+ k2, 0, k3, -k3/4 + k4/2], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
beta:=[[1, k1 + k2, -k3 - 2*k6, k3, k5/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr014b := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, -k2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, k2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1/2, -k2/2 + 2*k3, k2/2, 0], [0, -1, 0, 0, 0],
[0, 0, 1, 0, 1/2], [0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, k4, k2 - 2*k3, k2 + 2*k3 - 2*k5 + 2*k6, k5/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr015 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1/4 + k2, k1/4 + k2, -2*k6, k3/2 + k6/2], [0, 0, -1, 0, 0],
[0, -1, 0, 0, 0], [0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, k4, k1 + 2*k2 - k4, -2*k6, k5/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr018 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1 + 2*k2 - 2*k3 + 2*k4, k1/2 - 2*k3, 0,
k2/2 - (-k1 + 2*k2 - 2*k3 + 2*k4)/4], [0, -1, 0, 0, 1/2],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1/2, 2*k3, 0, 0], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr019 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1/2, 0, k1/2 + 2*k2, 0], [0, -1, 0, 0, 1/2],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, 3*k1 + 2*k2 - 2*k3 + 2*k4, 0, -k1 - 2*k2, k3/2],
[0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr019b := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1 + 2*k2 - 2*k3 + 2*k4, k1/2 - 2*k3, 0,
k2/2 - (-k1 + 2*k2 - 2*k3 + 2*k4)/4], [0, -1, 0, 0, 1/2],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1/2, 2*k3, 0, 0], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr019c := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, k1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, 0, -k1/2, 2*k2, -(- (3*k1)/2 - k4)/2], [0, -1, 0, 0, 1/2],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, 0, 2*k3, k1/2, 0], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr026 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1/2, 0, 2*k2, 0], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, 0, k3, 0, k4/2], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr027 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, 0, k4/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k2, 0, 2*k5, 0], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr029 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1/2, 0, 2*k2, 0], [0, -1, 0, 0, 0], [0, 0, -1, 0, 1/2],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, k1/2, 2*k2 - 2*k3 + 2*k4, 0, k3/2], [0, 1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr029b := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1/2 + k2, 2*k3 - 2*k4 + 2*k5, 0, k3/2 - (2*k3 - 2*k4 + 2*k5)/4],
[0, -1, 0, 0, 0], [0, 0, -1, 0, 1/2], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1 - k2, 0, 2*k4, 0], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr029c := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -2*k1, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 2*k1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, 0, -k1, -k1 + 2*(k1 + k2), -k1], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 1/2], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, -k3, -k1, 2*(k1 + k2), -k4/2], [0, 1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr030 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1/2 + k2, 2*k3 + 2*k5, 0, k3/2 - (2*k3 + 2*k5)/4],
[0, -1, 0, 0, 0], [0, 0, -1, 0, 1/2], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1 - k2, 0, 2*k4, 0], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr031 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1/2, 0, 2*k2, 0], [0, -1, 0, 0, 0], [0, 0, -1, 0, 1/2],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, 0, -2*k3 + 2*k4, 0, k3/2], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr032 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1/2 - 2*k4, -3*k1 + 2*k2 - 2*k4 + 2*k5, 0,
k2/2 - (-3*k1 + 2*k2 - 2*k4 + 2*k5)/4], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 1/2], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, 2*k4, k1/2, -k3, 0], [0, 1, 0, 0, 1/2], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr033 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1/2, 0, 2*k2, 0], [0, -1, 0, 0, 0], [0, 0, -1, 0, 1/2],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, 0, -k1 - 2*k3 + 2*k4, -k1/2, k3/2], [0, 1, 0, 0, 1/2],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr033b := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1/2 - 2*k4, -3*k1 + 2*k2 + k3 - 2*k4 + 2*k5, 0,
k2/2 - (-3*k1 + 2*k2 + k3 - 2*k4 + 2*k5)/4], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 1/2], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, 2*k4, k1/2, -k3, 0], [0, 1, 0, 0, 1/2], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr033c := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, k1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, 0, -k1/2, k1/2 + 2*k2, k2/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 1/2], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, 2*k3, 0, k1 + 2*k2, -k4/2], [0, 1, 0, 0, 1/2], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr034 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1/2 + k2, -2*k1 + k2 + 2*k3 + 2*k5, 0,
k3/2 - (-2*k1 + k2 + 2*k3 + 2*k5)/4], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 1/2], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1 - k2, k1/2, k1 + k2 + 2*k4, 0], [0, 1, 0, 0, 1/2],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr036 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, k1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1/2, -k1/2, 2*k2, 0], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, k3, -k3, 0, k4/2], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr037 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k2 + 2*k4, 0, k3/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k4, -k4, 2*k5, 0], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr041 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, k1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k2, k1 - 2*k5, -k2/2 + k3/2], [0, 0, -1, 0, 1/2],
[0, -1, 0, 0, 1/2], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, k1/2 - k4, (-3*k1)/2 + k4, 2*k5, 0], [0, 0, -1, 0, 0],
[0, -1, 0, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr043 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
c :=[[1, k1/2, -k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1/4 + k2 + 2*k3 + 2*k5, k1/4 + k2, 2*k2 + 2*k3 + 2*k5,
k3/2 - (2*k2 + 2*k3 + 2*k5)/4], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0],
[0, -1, -1, -1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1/4 + k2 + 2*k4, 2*k4, -k1/4 - k2, 0], [0, 0, 0, -1, 0],
[0, 1, 1, 1, 1/2], [0, -1, 0, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr045 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a:=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b:=[[1, 0, 0, k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c:=[[1, k1/2, -k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d:=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:=[[1,k2, -k2, k2 - 2*k4 - 2*k5, k3/2], [0, 0, 1, -1, 0], [0, 1, 0, -1, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
beta:=[[-1, -k4, k4 + 2*k5, k4, 0], [0, 0, 1, -1, 1/2], [0, 0, 1, 0, 1/2],
[0, -1, 1, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr055 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta, gamma;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1 - 2*k2 + 2*k3 - 2*k4, -k1 - 2*k3, 0, k2/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1/2, k1/2 + 2*k3, 0, 0], [0, -1, 0, 0, 1/2],
[0, 0, 1, 0, 1/2], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
gamma :=[[1, -k1 - 2*k2 + 2*k3 - 2*k4, -k1 - 2*k3, k5,
(2*k1 + k2 + k4 + k6)/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta, gamma] , IdentityMat(5) );
end;
ACDim4Nr056 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta, gamma;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1/2 + 2*k2 - 2*k3 + 2*(k1 - k2 + 2*k3 - k4), k1/2 - 2*k3, 0,
k2/2 - (k1 - 2*k3)/4 - (-k1 + 2*k2 - 2*k3 + 2*(k1 - k2 + 2*k3 - k4))/4],
[0, -1, 0, 0, 1/2], [0, 0, -1, 0, 1/2], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1/2, 2*k3, 0, 0], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
gamma :=[[1, 2*k2 - 2*k3 + 2*(k1 - k2 + 2*k3 - k4), -2*k3,
2*k3 + 2*k5 - 2*k6, k6/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta, gamma] , IdentityMat(5) );
end;
ACDim4Nr058 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta, gamma;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1 - 2*k2 + 2*k3 - 2*k4, -k1 - 2*k3, 0, k2/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1/2, k1/2 + 2*k3, 0, 0], [0, -1, 0, 0, 1/2],
[0, 0, 1, 0, 1/2], [0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
gamma :=[[1, -k1 - 2*k2 + 2*k3 - 2*k4, -k1 - 2*k3,
4*k1 + 2*k2 + 2*k4 + 2*k5 - 2*k6, k6/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta, gamma] , IdentityMat(5) );
end;
ACDim4Nr060 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta, gamma;
a :=[[1, 0, -2*k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 2*k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -2*(k1 - k2 + k3 - k4), k1 - 2*k3, 0,
k2/2 + (k1 - k2 + k3 - k4)/2], [0, -1, 0, 0, 1/2], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1, 2*k3, 0, 0], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
gamma :=[[1, -2*(k1 - k2 + k3 - k4), -2*k3, 2*k3 + 2*k5 - 2*k6, k6/2],
[0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta, gamma] , IdentityMat(5) );
end;
ACDim4Nr061 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta, gamma;
a :=[[1, 0, 0, -2*k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 2*k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1, 0, k1 + 2*k2, 0], [0, -1, 0, 0, 1/2], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, 7*k1 + 2*k2 - 2*k3 + 2*k4, 0, -2*k1 - 2*k2,
-(-2*k1 - 2*k2)/4 + k3/2], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
gamma :=[[1, 8*k1 + 2*k2 - 2*k3 + 2*k4, 2*k1 + 2*k2 - 2*k6, -2*k1 - 2*k2,
(-k1 + k3 - k4 + k5)/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta, gamma] , IdentityMat(5) );
end;
ACDim4Nr061b := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta, gamma;
a :=[[1, 0, -2*k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 2*k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -2*k3 - 2*k5 - 2*k6 + 2*(-k1 + k2 + k4 + k5 + k6), k1 - 2*k3, 0,
k2/2 - (-2*k3 - 2*k5 - 2*k6 + 2*(-k1 + k2 + k4 + k5 + k6))/4],
[0, -1, 0, 0, 1/2], [0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1, 2*k3, 0, 0], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
gamma :=[[1, -2*k3 - 2*k5 - 2*k6 + 2*(-k1 + k2 + k4 + k5 + k6), -2*k3,
2*k3 + 2*k6 - 2*(-k1 + k2 + k4 + k5 + k6), (-k1 + k2 + k4 + k5 + k6)/2],
[0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta, gamma] , IdentityMat(5) );
end;
ACDim4Nr061c := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta, gamma;
a :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, -2*k1, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 2*k1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, 0, -k1, 2*k2, 2*k1 - k2/2], [0, -1, 0, 0, 1/2],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, 0, -3*k1 - 2*k2 - 2*k6 + 2*(k1 + k2 + k3 + k6), k1, -k4/2],
[0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2], [0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
gamma :=[[1, 2*k2 + 2*k5 - 2*(k1 + k2 + k3 + k6),
4*k1 + 2*k2 + 2*k6 - 2*(k1 + k2 + k3 + k6), -2*k2, (k1 + k2 + k3 + k6)/2],
[0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta, gamma] , IdentityMat(5) );
end;
ACDim4Nr062 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta, gamma;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[-1, -k1/2, 0, k1/2 + 2*k2, 0], [0, -1, 0, 0, 1/2],
[0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, 3*k1 + 2*k2 - 2*k3 + 2*k4, 0, -k1 - 2*k2, k3/2],
[0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
gamma :=[[1, 3*k1 + 2*k2 - 2*k3 + 2*k4, -2*k6, -k1 - 2*k2, (k3 - k4 + k5)/2],
[0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta, gamma] , IdentityMat(5) );
end;
ACDim4Nr075 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, 0, k4/4], [0, 0, -1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr076 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, 0, k4/4], [0, 0, -1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 1/4], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr077 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, 0, k4/4], [0, 0, -1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr079 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, -k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k2, k3, k4/4], [0, 0, 1, 0, 0], [0, 0, 1, -1, 0],
[0, -1, 1, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr080 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, -k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1/4 + k2, k1/4 - k2, k3, k1/16 - k2/4 + k3/4 + k4/4],
[0, 0, 1, 0, 1/2], [0, 0, 1, -1, 0], [0, -1, 1, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr081 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, k4, k5/4], [0, 0, 1, 0, 0], [0, -1, 0, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr082 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, -k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, k4, k5/4], [0, 0, -1, 0, 0], [0, 0, -1, 1, 0],
[0, 1, -1, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr083 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, 0, k4/4], [0, 0, -1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[1, k2 + k3, -k2 + k3, k5, k6/2], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr084 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, 0, k4/4], [0, 0, -1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, k2 + k3, -k2 + k3, -2*k6, k5/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr085 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k2 - 2*k6, 0,
-k1/8 + k2/4 + k3/4 - (-k1 + k2 - 2*k6)/4], [0, 0, -1, 0, 0],
[0, 1, 0, 0, 1/2], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[1, 2*k2 - 2*k6, -2*k6, k4, k5/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr086 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k1/2 + k3, 0, -k1/8 + k2/4 - k3/4 + k4/4], [0, 0, -1, 0, 0],
[0, 1, 0, 0, 1/2], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, k1 + k2 + k3, k1 - k2 + k3, -k1 + k2 - k3 - 2*k6, k5/2],
[0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr087 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, -k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k2, k3, k4/4], [0, 0, 1, 0, 0], [0, 0, 1, -1, 0],
[0, -1, 1, 0, 0], [0, 0, 0, 0, 1]];
beta :=[[1, k5, -2*k2 + 2*k3 + k5, 2*k2 - k5, k6/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr088 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1/2, -k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k2, -k1/4 + k3, -(-k1/4 + k2 + k3 - k4)/4], [0, 0, 1, 0, 0],
[0, 0, 1, -1, 0], [0, -1, 1, 0, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, 2*k2 + 2*k6, -k1 + 2*k3 + 2*k6, -2*k6, k5/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr103 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, 0, k4/4], [0, 0, -1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k2 - k3, 0, 2*k5, 0], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr104 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1/2 + k2, -k1 + k2 + 2*k3 + 2*k5, 0,
-k1/8 - k2/4 + k3/4 - (-k1 + k2 + 2*k3 + 2*k5)/4], [0, 0, -1, 0, 1/2],
[0, 1, 0, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -2*k2 - 2*k3 - 2*k5, k1/2, 2*k2 + 2*k3 + 2*k4 + 2*k5, 0],
[0, 1, 0, 0, 1/2], [0, 0, -1, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr106 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1/2 + k2, -k1 - k2 - 2*k5, 0,
-k1/8 - k2/4 + k3/4 - (-k1 - k2 - 2*k5)/4], [0, 0, -1, 0, 1/2],
[0, 1, 0, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, 2*k5, k1/2, -2*k4, 0], [0, 1, 0, 0, 1/2], [0, 0, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr110 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, 0, -k1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, 0, 0, k1, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, k1, -k1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, 4*k1 - 4*k2 + 2*k3 - 3*k4 + 2*k5,
-4*k1 + 4*k2 - 2*k3 + 3*k4 - 2*k5, -k1/2 + k2,
-(-k1/2 + k2 - k3 - (3*(-4*k1 + 4*k2 - 2*k3 + 3*k4 - 2*k5))/2 -
(4*k1 - 4*k2 + 2*k3 - 3*k4 + 2*k5)/2)/4], [0, 0, 1, 0, 0],
[0, 0, 1, -1, 0], [0, -1, 1, 0, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, -k4, 2*(k1 - k2 - k4) + k4, k4, 0], [0, 0, 1, -1, 1/2],
[0, 0, 1, 0, 1/2], [0, -1, 1, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr114 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, -k1/2 + k2, k1 - k2 - 2*k4, 2*k1 - 2*k2 + 2*k3 - 2*k4 + 2*k5,
-(k1/2 + k3 + 2*k5)/4], [0, 0, 1, 0, 1/2], [0, -1, 0, 0, 0],
[0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1/2, 2*k4, 0, 0], [0, -1, 0, 0, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr143 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a:= [[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b:= [[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c:= [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[1, k2, -k1/2 + k3, 0, k4/3], [0, 0, -1, 0, 0], [0, 1, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr144 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, 0, k4/3], [0, 0, -1, 0, 0], [0, 1, -1, 0, 0],
[0, 0, 0, 1, 1/3], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr146 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, -k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
c :=[[1, -k1/2, k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, -k2 - k3, k4/3], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0],
[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr147 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, k4, k5/6], [0, 0, 1, 0, 0], [0, -1, 1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr148 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, -k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
c :=[[1, -k1/2, k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, k3, k4, k5/6], [0, 0, 0, -1, 0], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr158 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, 0, k4/3], [0, 0, -1, 0, 0], [0, 1, -1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k2, k2, 2*k5, 0], [0, 0, -1, 0, 0], [0, -1, 0, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr159 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1 - 2*k2 + 3*k4, -k1/2 + k2, 0, k3/3], [0, 0, -1, 0, 0],
[0, 1, -1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k4, -k4, 2*k5, 0], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr161 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, k1/2, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, -k1/2, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
c :=[[1, -k1/2, k1/2, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k1/4 + k2, k3, -k1/4 - k2 - k3,
-((5*k1)/8 + k2 + (-k2 - k3)/2 + (3*k3)/2 - k4)/3], [0, 0, 0, 1, 1/2],
[0, 1, 0, 0, 1/2], [0, 0, 1, 0, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1/4 - k2 - 2*k3 + 2*k5, (-3*k1)/4 - k2 - 2*k3 + 2*k5, 2*k5,
0], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr168 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, 0, k4/6], [0, 0, 1, 0, 0], [0, -1, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr169 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, 0, k4/6], [0, 0, 1, 0, 0], [0, -1, 1, 0, 0],
[0, 0, 0, 1, 5/6], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr172 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, 0, k4/6], [0, 0, 1, 0, 0], [0, -1, 1, 0, 0],
[0, 0, 0, 1, 1/3], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr173 := function ( k1, k2, k3, k4)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, 0, k4/6], [0, 0, 1, 0, 0], [0, -1, 1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr174 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, k4, k5/6], [0, 0, -1, 0, 0], [0, 1, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4Nr175 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, 0, k4/6], [0, 0, 1, 0, 0], [0, -1, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[1, k1 - 2*k3, -k1 + 2*k2 + 2*k3, k5, k6/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr176 := function ( k1, k2, k3, k4, k5, k6)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, 0, k4/6], [0, 0, 1, 0, 0], [0, -1, 1, 0, 0],
[0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
beta :=[[1, k1 - 2*k3, -k1 + 2*k2 + 2*k3, 2*k6, k5/2], [0, -1, 0, 0, 0],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4Nr184 := function ( k1, k2, k3, k4, k5)
local a, b, c, d, alfa, beta;
a :=[[1, 0, -k1/2, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b :=[[1, k1/2, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
c :=[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]];
d :=[[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa :=[[1, k2, -k1/2 + k3, 0, k4/6], [0, 0, 1, 0, 0], [0, -1, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
beta :=[[-1, -k1 + k2 + 2*k3, k1 - k2 - 2*k3, 2*k5, 0], [0, 0, -1, 0, 0],
[0, -1, 0, 0, 0], [0, 0, 0, 1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a, b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4NrB1 := function ( k, k1, k2, k3)
local a, b, c, d;
a:= [[1, (-2*k2)/3, 0, -k1/2 - (2*k*k3)/3 + (2*k*(k2 + k3))/3, 0],
[0, 1, 0, -k/2, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (-2*k3)/3, k1/2 - (2*k*k3)/3, 0, 0], [0, 1, k/2, 0, 0],
[0, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, k2/3, k3/3, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
return Group( [a , b, c, d] , IdentityMat(5) );
end;
ACDim4NrB2 := function ( k, k1, k2, k3)
local a, b, c, d, alfa;
a:= [[1, (-4*k1)/3, 0, (2*k*k1)/3 + (2*k*k2)/3 - 2*k*k3 +
(-4*k*k1 - (16*k*k2)/3 - 2*k*k3 + 2*(2*k*k1 + 2*k*k2 + 2*k*k3))/2, 0],
[0, 1, 0, -k, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (-4*k2)/3, (-4*k*k1 - (16*k*k2)/3 - 2*k*k3 +
2*(2*k*k1 + 2*k*k2 + 2*k*k3))/2, 0, 0], [0, 1, k, 0, 0],
[0, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, (2*k1)/3, (2*k2)/3, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[-1, 2*k3, -k1/3, -k2/3, 0], [0, 1, 0, 0, 1/2], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4NrB3c := function (l, k1, k2, k3, k4)
local a, b, c, d, alfa;
a:= [[1, 0, 0, (k1*l)/3 + k3*l + ((-8*k1*l)/3 + k3*l + 2*(k1*l - k3*l))/2,
0], [0, 1, 0, -l, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (-2*k1)/3, ((-8*k1*l)/3 + k3*l + 2*(k1*l - k3*l))/2, -k2, 0],
[0, 1, l, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, 0, k1/3, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[1, k3, 0, 0, k4/2], [0, -1, 0, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4NrB3b := function (l, k1, k2, k3, k4)
local a, b, c, d, alfa;
a:= [[1, 0, 0, k1 + (2*k1*l)/3 - 2*k2*l + ((2*k1)/3 - (16*k1*l)/3 - 2*k2*l +
2*(-k1 + 2*k1*l + 2*k2*l))/2, 0], [0, 1, 0, -l, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (-4*k1)/3, ((2*k1)/3 - (16*k1*l)/3 - 2*k2*l +
2*(-k1 + 2*k1*l + 2*k2*l))/2, -k3, 0], [0, 1, l, 0, 0],
[0, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, 0, (2*k1)/3, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[1, -2*k2, -k2, 0, k4/2], [0, -1, -1, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4NrB3 := function (l, k1, k2, k3, k4)
local a, b, c, d, alfa;
a:= [[1, 0, 0, (2*k1)/3 + (k1*l)/3 + (-k1 - (2*k1*l)/3)/2, 0],
[0, 1, 0, (-1 - 2*l)/2, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
b:= [[1, (-2*k1)/3, (-k1 - (2*k1*l)/3)/2, -k3, 0], [0, 1, (1 + 2*l)/2, 0, 0],
[0, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, 0, k1/3, -k2], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[1, 0, 0, 0, k4/2], [0, -1, -1, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4NrB4 := function (k, k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a:= [[1, 0, 0, (k*k1)/3 + k*k3 + ((-2*k*k1)/3 - k*k3)/2, -k4],
[0, 1, 0, -k, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (-2*k1)/3, ((-2*k*k1)/3 - k*k3)/2, -(k*k1)/2 - k2 + (3*k*k3)/4, 0],
[0, 1, k, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, 0, k1/3, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[1, k3, 0, 0, 0], [0, -1, 0, k/2, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4NrB4b := function (k, k1, k2, k3)
local a, b, c, d, alfa;
a:= [[1, (4*k1)/3, 0, (-5*k*k1)/3 + 2*k2 - 2*((-2*k*k1)/3 + k2), -k3],
[0, 1, 0, -k, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, 0, -2*((-2*k*k1)/3 + k2), 0, 0], [0, 1, k, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, (-2*k1)/3, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[-1, (-2*k1)/3, 0, 0, 0], [0, -1, 0, k/2, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4NrB5 := function (l, k1, k2, k3, k4)
local a, b, c, d, alfa;
a:= [[1, (-2*k1)/3, 0, k3*l, k2], [0, 1, 0, -l, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (2*k1)/3, 0, 0, 0], [0, 1, l, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, k1/3, -k1/3, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[1, k3, 0, 0, k4/2], [0, -1, 0, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4NrB5b := function (l, k1, k2, k3, k4)
local a, b, c, d, alfa;
a:= [[1, (-2*k1)/3, 0, k3*(1 + 2*l), k2], [0, 1, 0, (-1 - 2*l)/2, 0],
[0, 0, 1, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (2*k1)/3, 0, 0, 0], [0, 1, (1 + 2*l)/2, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, k1/3, -k1/3, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[1, 2*k3, 0, 0, k4/2], [0, -1, 0, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa] , IdentityMat(5) );
end;
ACDim4NrB7 := function (l, k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a:= [[1, 0, 0, (4*k1*l)/3 - 4*k2*l + ((-8*k1*l)/3 + 4*k2*l)/2, 0],
[0, 1, 0, -2*l, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (-4*k1)/3, ((-8*k1*l)/3 + 4*k2*l)/2, 0, (5*k1)/6 - 2*k2 +
((-2*k1)/3 + 2*k2)/2 - k4], [0, 1, 2*l, 0, 0], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, 0, (2*k1)/3, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[-1, 2*k2, 0, -k1/3, 0], [0, 1, 0, 0, 1/2], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
beta:= [[1, (2*k1)/3 - 2*k2, -(((2*k1)/3 - 2*k2)*l)/2, 2*k3, 0],
[0, -1, l, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4NrB7b := function (l, k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a:= [[1, 0, 0, (4*k1*(1 + 2*l))/3 - 2*k2*(1 + 2*l) +
((-8*k1*(1 + 2*l))/3 + 2*k2*(1 + 2*l))/2, 0], [0, 1, 0, -1 - 2*l, 0],
[0, 0, 1, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (-8*k1)/3, ((-8*k1*(1 + 2*l))/3 + 2*k2*(1 + 2*l))/2, 0,
(5*k1)/3 - 2*k2 + ((-4*k1)/3 + 2*k2)/2 - k4], [0, 1, 1 + 2*l, 0, 0],
[0, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, 0, (4*k1)/3, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[-1, 2*k2, 0, (-2*k1)/3, 0], [0, 1, 0, 0, 1/2], [0, 0, -1, 0, 0],
[0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
beta:= [[1, (4*k1)/3 - 2*k2, -(((4*k1)/3 - 2*k2)*(1 + 2*l))/4, 2*k3, 0],
[0, -1, (1 + 2*l)/2, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, -1, 1/2],
[0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4NrB8 := function (k, k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a:= [[1, 0, 0, -2*k*(k1 - 2*k2) - 8*k*k2 + (8*k*(2*(k1 - 2*k2) + 4*k2))/3 +
(-4*k*k2 - (14*k*(2*(k1 - 2*k2) + 4*k2))/3 +
2*(2*k*(k1 - 2*k2) + 8*k*k2))/2,
(k*(k1 + (-2*(k1 - 2*k2) - 4*k2)/6 - 2*k2))/4 - k3 +
(k1 + (17*k*(k1 - 2*k2))/3 - 3*k2 + (34*k*k2)/3 + (-k1 + 2*k2)/2 +
(-2*k*(k1 - 2*k2) - 8*k*k2)/2 + (k1 - 2*k*(k1 - 2*k2) - 6*k*k2)/2 +
2*k3 - 2*k4)/2], [0, 1, 0, -2*k, 0], [0, 0, 1, 0, 1],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, (-2*(2*(k1 - 2*k2) + 4*k2))/3,
(-4*k*k2 - (14*k*(2*(k1 - 2*k2) + 4*k2))/3 +
2*(2*k*(k1 - 2*k2) + 8*k*k2))/2, k1 + (-2*(k1 - 2*k2) - 4*k2)/3 -
2*k*(k1 - 2*k2) - 2*k*k2 + (4*k*(2*(k1 - 2*k2) + 4*k2))/3 +
((-2*(k1 - 2*k2) - 4*k2)/3 - 4*k*k2)/2, 0], [0, 1, 2*k, 0, 0],
[0, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, 0, (2*(k1 - 2*k2) + 4*k2)/3, 0], [0, 1, 0, 0, 1],
[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[-1, 2*k2, 2*k*k2, ((-2*(k1 - 2*k2) - 4*k2)/3 - 4*k*k2)/2, 0],
[0, 1, 2*k, -2*k, 1/2], [0, 0, -1, 0, 0], [0, 0, 0, -1, 0],
[0, 0, 0, 0, 1]];
beta:= [[1, k1 + (-2*(k1 - 2*k2) - 4*k2)/6 - 2*k2,
(k*(k1 + (-2*(k1 - 2*k2) - 4*k2)/6 - 2*k2))/2,
k1 + (17*k*(k1 - 2*k2))/3 - 3*k2 + (34*k*k2)/3 + (-k1 + 2*k2)/2 +
(-2*k*(k1 - 2*k2) - 8*k*k2)/2 + (k1 - 2*k*(k1 - 2*k2) - 6*k*k2)/2 +
2*k3 - 2*k4, 0], [0, -1, -k, k, 0], [0, 0, 1, 0, 1/2],
[0, 0, 0, -1, 1/2], [0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa, beta] , IdentityMat(5) );
end;
ACDim4NrB8b := function (k, k1, k2, k3, k4)
local a, b, c, d, alfa, beta;
a:= [[1, (4*k1)/3, (-2*k1)/3 + (4*k*k1)/3 +
((-16*k*k1)/3 + 8*k*k2 - 2*(-2*k*k1 + 4*k*k2))/2 +
2*(k1/3 - (2*k*k1)/3 + ((16*k*k1)/3 - 8*k*k2 +
2*(-2*k*k1 + 4*k*k2))/ 2), (-10*k*k1)/3 + ((16*k*k1)/3 - 8*k*k2 +
2*(-2*k*k1 + 4*k*k2))/2,
((8*k*k1)/3 - 2*k*k2 + (-2*k*k1 + 4*k*k2)/2 +
((-16*k*k1)/3 + 8*k*k2 - 2*(-2*k*k1 + 4*k*k2))/2)/2 - k3],
[0, 1, 0, -2*k, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
b:= [[1, 0, ((16*k*k1)/3 - 8*k*k2 + 2*(-2*k*k1 + 4*k*k2))/2, 0,
(-2*k1)/3 - (2*k*k1)/3 + k2 + 4*k*k2 + (2*k*k1 - 4*k*k2)/2 +
((-16*k*k1)/3 + 8*k*k2 - 2*(-2*k*k1 + 4*k*k2))/4 +
(k1/3 - (2*k*k1)/3 + ((16*k*k1)/3 - 8*k*k2 + 2*(-2*k*k1 + 4*k*k2))/2)/ 2
+ ((-8*k*k1)/3 + 2*k*k2 + (2*k*k1 - 4*k*k2)/2 +
((16*k*k1)/3 - 8*k*k2 + 2*(-2*k*k1 + 4*k*k2))/2)/2 + k3 + k4],
[0, 1, 2*k, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 1]];
c:= [[1, 0, (-2*k1)/3, 0, -k2], [0, 1, 0, 0, 1], [0, 0, 1, 0, 0],
[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]];
d:= [[1, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]];
alfa:= [[-1, 0, k1/3 - (2*k*k1)/3 + ((16*k*k1)/3 - 8*k*k2 +
2*(-2*k*k1 + 4*k*k2))/2, 0, 0], [0, 1, 2*k, -2*k, 1/2],
[0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 1]];
beta:= [[-1, (-2*k1)/3, 0, (8*k*k1)/3 - 2*k*k2 + (-2*k*k1 + 4*k*k2)/2 +
((-16*k*k1)/3 + 8*k*k2 - 2*(-2*k*k1 + 4*k*k2))/2, 0],
[0, -1, -k, k, 0], [0, 0, 1, 0, 1/2], [0, 0, 0, -1, 1/2],
[0, 0, 0, 0, 1]];
return Group( [a , b, c, d, alfa, beta] , IdentityMat(5) );
end;
#############################################################################
##
#V ACDim4Funcs
#V ACDim4Param
#V ACDim4Types
##
#############################################################################
##
## some small helpers
##
ACDim4Funcs := [ ACDim4Nr001, ACDim4Nr002, ACDim4Nr003, ACDim4Nr004,
ACDim4Nr004b, ACDim4Nr005, ACDim4Nr006, ACDim4Nr007, ACDim4Nr007b,
ACDim4Nr008, ACDim4Nr009, ACDim4Nr009b, ACDim4Nr010, ACDim4Nr011,
ACDim4Nr012, ACDim4Nr013, ACDim4Nr014,
ACDim4Nr014b, ACDim4Nr015, ACDim4Nr018, ACDim4Nr019, ACDim4Nr019b,
ACDim4Nr019c, ACDim4Nr026, ACDim4Nr027, ACDim4Nr029, ACDim4Nr029b,
ACDim4Nr029c, ACDim4Nr030, ACDim4Nr031, ACDim4Nr032, ACDim4Nr033,
ACDim4Nr033b, ACDim4Nr033c, ACDim4Nr034, ACDim4Nr036, ACDim4Nr037,
ACDim4Nr041, ACDim4Nr043, ACDim4Nr045, ACDim4Nr055, ACDim4Nr056,
ACDim4Nr058, ACDim4Nr060, ACDim4Nr061, ACDim4Nr061b, ACDim4Nr061c,
ACDim4Nr062, ACDim4Nr075, ACDim4Nr076, ACDim4Nr077, ACDim4Nr079,
ACDim4Nr080, ACDim4Nr081, ACDim4Nr082, ACDim4Nr083, ACDim4Nr084, ACDim4Nr085,
ACDim4Nr086, ACDim4Nr087, ACDim4Nr088, ACDim4Nr103, ACDim4Nr104, ACDim4Nr106,
ACDim4Nr110, ACDim4Nr114, ACDim4Nr143, ACDim4Nr144, ACDim4Nr146, ACDim4Nr147,
ACDim4Nr148, ACDim4Nr158, ACDim4Nr159, ACDim4Nr161, ACDim4Nr168, ACDim4Nr169,
ACDim4Nr172, ACDim4Nr173, ACDim4Nr174, ACDim4Nr175, ACDim4Nr176, ACDim4Nr184,
ACDim4NrB1, ACDim4NrB2, ACDim4NrB3c, ACDim4NrB3b, ACDim4NrB3, ACDim4NrB4,
ACDim4NrB4b, ACDim4NrB5, ACDim4NrB5b, ACDim4NrB7, ACDim4NrB7b, ACDim4NrB8,
ACDim4NrB8b ];
MakeReadOnlyGlobal( "ACDim4Funcs" );
#############################################################################
ACDim4Types := [
"001", "002", "003", "004", "004b", "005", "006", "007", "007b", "008",
"009", "009b", "010", "011", "012", "013", "014", "014b", "015", "018",
"019", "019b", "019c", "026", "027", "029", "029b", "029c", "030", "031",
"032", "033", "033b", "033c", "034", "036", "037", "041", "043", "045",
"055", "056", "058", "060", "061", "061b", "061c", "062", "075", "076",
"077", "079", "080", "081", "082", "083", "084", "085", "086", "087", "088",
"103", "104", "106", "110", "114", "143", "144", "146", "147", "148", "158",
"159", "161", "168", "169", "172", "173", "174", "175", "176", "184", "B1",
"B2", "B3c", "B3b", "B3", "B4", "B4b", "B5", "B5b", "B7", "B7b", "B8", "B8b"
];
MakeReadOnlyGlobal( "ACDim4Types" );
#############################################################################
ACDim4Param :=
[ 3, 7, 4, 4, 3, 4, 3, 3, 4, 3, 3, 4, 6, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 4, 5,
4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4,
4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 4, 4, 4, 5, 5, 5, 5, 5, 4,
4, 4, 4, 5, 6, 6, 5, 4, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5 ];
MakeReadOnlyGlobal( "ACDim4Param" );
aclib-1.3.2/gap/betti.gi 0000664 0003717 0003717 00000011056 13627151544 016213 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W betti.gi Karel Dekimpe
#W Bettina Eick
##
#############################################################################
##
## The following functions can be used to determine Betti-numbers of a
## torsion-free polycyclic group given by a pcp presentation. All
## Betti-numbers can be obtained if G has Hirsch length at most 5.
##
## The Betti-numbers B(G,m) are defined as the ranks of H_m(G,Z) for the
## trivial G-module Z. If M is the orientation G-module, then we can also
## characterise B(G,m) for n >= m >= n-2 as the ranks of H^n-m(G,M).
## Further, the alternating sum of all Betti-numbers is 0 using the
## Euler characteristic.
##
#############################################################################
##
#F OrientationModule( G )
##
InstallMethod( OrientationModule, "for pcp groups", true, [IsPcpGroup], 0,
function( G )
local pcps, gens, mats, acts, dets, i, pcp;
pcps := PcpsOfEfaSeries( G );
pcps := Filtered( pcps, x -> RelativeOrdersOfPcp(x)[1] = 0 );
gens := Igs(G);
mats := List( gens, x -> IdentityMat( 1 ) );
for pcp in pcps do
acts := LinearActionOnPcp( gens, pcp );
dets := List( acts, x -> Determinant( x ) );
for i in [1..Length(mats)] do
mats[i] := dets[i] * mats[i];
od;
od;
return mats;
end );
#############################################################################
##
#F IsOrientedMatGroup( G )
##
IsOrientedMatGroup := function( G )
return ForAll( GeneratorsOfGroup(G), x -> Determinant(x) = 1 );
end;
#############################################################################
##
#F BettiNumber( G, m )
##
BettiNumberPcpGroup := function(G,m)
local n, pcp, mats, CR, one, two;
if not IsTorsionFree( G ) then
Print("the input group must be torsion-free \n");
return fail;
fi;
# catch the trivial case
if IsFinite(G) then
if m = 0 then
return 1;
else
return 0;
fi;
fi;
# the hirsch length
n := HirschLength( G );
if m < 0 or m > n then return 0; fi;
if m = 0 then return 1; fi;
if m = 1 then
pcp := Pcp( G, DerivedSubgroup(G) );
return Length( Filtered( RelativeOrdersOfPcp( pcp ),x -> x=0 ));
fi;
if m = n then
mats := OrientationModule( G );
if ForAny( mats, x -> x[1][1] = -1 ) then
return 0;
else
return 1;
fi;
fi;
if m = 2 then
mats := List( Pcp(G), x -> IdentityMat(1) );
CR := CRRecordByMats( G, mats );
two := TwoCohomologyCR( CR ).factor.rels;
return Length( Filtered( two, x -> x = 0 ) );
fi;
if m = n-1 then
mats := OrientationModule( G );
CR := CRRecordByMats( G, mats );
one := OneCohomologyCR( CR ).factor.rels;
return Length( Filtered( one, x -> x = 0 ) );
fi;
if m = n-2 then
mats := OrientationModule( G );
CR := CRRecordByMats( G, mats );
two := TwoCohomologyCR( CR ).factor.rels;
return Length( Filtered( two, x -> x = 0 ) );
fi;
Print("Betti-number is out of range for our methods \n");
return fail;
end;
InstallMethod( BettiNumber, "for torsion-free pcp groups", true,
[IsPcpGroup, IsInt], 0,
function(G, m)
if not IsTorsionFree(G) then TryNextMethod(); fi;
if m in [3..HirschLength(G)-3] then TryNextMethod(); fi;
return BettiNumberPcpGroup(G,m);
end);
#############################################################################
##
#F BettiNumbers( G )
##
InstallMethod( BettiNumbers, "for torsion-free pcp groups", true,
[IsPcpGroup], 0,
function( G )
local n, betti;
n := HirschLength( G );
if not IsTorsionFree( G ) or n > 6 then TryNextMethod(); fi;
# set up the Betti-numbers
betti := [1];
if n = 0 then return betti; fi;
betti[2] := BettiNumber( G, 1 );
if n = 1 then return betti; fi;
betti[3] := BettiNumber( G, 2 );
if n = 2 then return betti; fi;
betti[4] := betti[1] - betti[2] + betti[3];
if n = 3 then return betti; fi;
if n > 3 then
betti[5] := BettiNumber( G, 4 );
betti[4] := betti[4] + betti[5];
fi;
if n > 4 then
betti[6] := BettiNumber( G, 5 );
betti[4] := betti[4] - betti[6];
fi;
if n > 5 then
betti[7] := BettiNumber( G, 6 );
betti[4] := betti[4] + betti[7];
fi;
return betti;
end );
aclib-1.3.2/gap/matgrp3.gi 0000664 0003717 0003717 00000017621 13627151544 016465 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W matgrp3.gi Karel Dekimpe
#W Bettina Eick
##
## This file contains the 3-dimensional almost crystallographic groups
## as integral matrix groups. There are 17 types of groups.
##
ACDim3Nr01 := function ( k1 )
local a, b, c;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
return Group( [a , b, c] , IdentityMat(4) );
end;
ACDim3Nr02 := function ( k1, k2, k3, k4)
local a, b, c, alfa;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k2, k3, k4/2], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa] , IdentityMat(4) );
end;
ACDim3Nr03 := function ( k1, k2 )
local a, b, c, alfa;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[-1, -k2, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa] , IdentityMat(4) );
end;
ACDim3Nr04 := function ( k1, k2 )
local a, b, c, alfa;
a:=[[1, 0, -k1, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[-1, 2*k2, k1/2, 0], [0, 1, 0, 1/2], [0, 0, -1, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa] , IdentityMat(4) );
end;
ACDim3Nr05 := function ( k1, k2 )
local a, b, c, alfa;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[-1, -k2, -k2, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa ] , IdentityMat(4) );
end;
ACDim3Nr06 := function ( k1, k2, k3, k4)
local a, b, c, alfa, beta;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k2, k3, k4/2], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]];
beta:=[[-1, -k2, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa, beta] , IdentityMat(4) );
end;
ACDim3Nr07 := function ( k1, k2, k3, k4)
local a, b, c, alfa, beta;
a:=[[1, 0, -k1, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k2, -2*(k3 + k4), k3/2], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]];
beta:=[[-1, k1/2 - k2, 0, 0], [0, 1, 0, 0], [0, 0, -1, 1/2], [0, 0, 0, 1]];
return Group( [a , b, c, alfa, beta] , IdentityMat(4) );
end;
ACDim3Nr08 := function ( k1, k2, k3, k4)
local a, b, c, alfa, beta;
a:=[[1, 0, -k1, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k1 + 2*k3, -k1 + 2*k2 - 2*k3, k4/2], [0, -1, 0, 0], [0, 0, -1, 0],
[0, 0, 0, 1]];
beta:=[[-1, -k1/2 - 2*k3, k1/2, 0], [0, 1, 0, 1/2], [0, 0, -1, 1/2], [0, 0, 0, 1]];
return Group( [a , b, c, alfa, beta] , IdentityMat(4) );
end;
ACDim3Nr09 := function ( k1, k2, k3, k4)
local a, b, c, alfa, beta;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k2, -k2 + 2*k3, k4/2], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]];
beta:=[[-1, -k3, -k3, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa, beta] , IdentityMat(4) );
end;
ACDim3Nr10 := function ( k1, k2, k3, k4)
local a, b, c, alfa;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k2, k3, k4/4], [0, 0, -1, 0], [0, 1, 0, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa] , IdentityMat(4) );
end;
ACDim3Nr11 := function ( k1, k2, k3, k4)
local a, b, c, alfa, beta;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k2, k3, k4/4], [0, 0, -1, 0], [0, 1, 0, 0], [0, 0, 0, 1]];
beta:=[[-1, -k2 - k3, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa, beta] , IdentityMat(4) );
end;
ACDim3Nr12 := function ( k1, k2, k3, k4)
local a, b, c, alfa, beta;
a:=[[1, 0, -k1, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k2, k1 - k2 - 2*k3, k4/4], [0, 0, -1, 0], [0, 1, 0, 0], [0, 0, 0, 1]];
beta:=[[-1, -k1/2 + 2*k3, k1/2, 0], [0, 1, 0, 1/2], [0, 0, -1, 1/2], [0, 0, 0, 1]];
return Group( [a , b, c, alfa, beta] , IdentityMat(4) );
end;
ACDim3Nr13 := function ( k1, k2, k3, k4)
local a, b, c, alfa;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k2, -k1/2 + k3, k4/3], [0, 0, -1, 0], [0, 1, -1, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa] , IdentityMat(4) );
end;
ACDim3Nr14 := function ( k1, k2, k3, k4)
local a, b, c, alfa, beta;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k2, -k1/2 + k3, k4/3], [0, 0, -1, 0], [0, 1, -1, 0], [0, 0, 0, 1]];
beta:=[[-1, -k2 , k2 , 0], [0, 0, -1, 0], [0, -1, 0, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa, beta] , IdentityMat(4) );
end;
ACDim3Nr15 := function ( k1, k2, k3, k4)
local a, b, c, alfa, beta;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, k1 - 2*k3 + 3*k2, -k1/2 + k3, k4/3], [0, 0, -1, 0], [0, 1, -1, 0],
[0, 0, 0, 1]];
beta:=[[-1, -k2, -k2, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa, beta] , IdentityMat(4) );
end;
ACDim3Nr16 := function ( k1, k2, k3, k4)
local a, b, c, alfa;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, -k1/2 + k2, k3, k4/6], [0, 1, -1, 0], [0, 1, 0, 0],
[0, 0, 0, 1]];
return Group( [a , b, c, alfa] , IdentityMat(4) );
end;
ACDim3Nr17 := function ( k1, k2, k3, k4)
local a, b, c, alfa, beta;
a:=[[1, 0, -k1/2, 0], [0, 1, 0, 1], [0, 0, 1, 0], [0, 0, 0, 1]];
b:=[[1, k1/2, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]];
c:=[[1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
alfa:=[[1, -k1/2 + k2, k3, k4/6], [0, 1, -1, 0], [0, 1, 0, 0],
[0, 0, 0, 1]];
beta:=[[-1, -k3, -k3, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]];
return Group( [a , b, c, alfa, beta] , IdentityMat(4) );
end;
#############################################################################
##
#V ACDim3Funcs
#V ACDim3Param
##
ACDim3Funcs := [ ACDim3Nr01, ACDim3Nr02, ACDim3Nr03, ACDim3Nr04, ACDim3Nr05,
ACDim3Nr06, ACDim3Nr07, ACDim3Nr08, ACDim3Nr09, ACDim3Nr10, ACDim3Nr11,
ACDim3Nr12, ACDim3Nr13, ACDim3Nr14, ACDim3Nr15, ACDim3Nr16, ACDim3Nr17 ];
MakeReadOnlyGlobal( "ACDim3Funcs" );
ACDim3Param := [ 1,4,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4 ];
MakeReadOnlyGlobal( "ACDim3Param" );
aclib-1.3.2/gap/crystgrp.gi 0000664 0003717 0003717 00000264527 13627151544 016776 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W In this file we prepare functions to loop over the 2,3 or 4 dimensional
#W crystallographic groups as matrix groups or pcp groups.
##
cryst4params:=
[ [ 1, 1, 1, 1 ], [ 1, 2, 1, 1 ], [ 2, 1, 1, 1 ], [ 2, 1, 1, 2 ],
[ 2, 1, 2, 1 ], [ 2, 1, 2, 2 ], [ 2, 2, 1, 1 ], [ 2, 2, 1, 2 ],
[ 2, 2, 2, 1 ], [ 2, 3, 1, 1 ], [ 2, 3, 1, 2 ], [ 2, 3, 1, 3 ],
[ 2, 3, 1, 4 ], [ 2, 3, 2, 1 ], [ 2, 3, 2, 2 ], [ 3, 1, 1, 1 ],
[ 3, 1, 1, 2 ], [ 3, 1, 2, 1 ], [ 3, 1, 2, 2 ], [ 3, 1, 3, 1 ],
[ 3, 2, 1, 1 ], [ 3, 2, 1, 2 ], [ 3, 2, 1, 3 ], [ 3, 2, 2, 1 ],
[ 3, 2, 2, 2 ], [ 3, 2, 2, 3 ], [ 3, 2, 3, 1 ], [ 4, 1, 1, 1 ],
[ 4, 1, 1, 2 ], [ 4, 1, 1, 3 ], [ 4, 1, 1, 4 ], [ 4, 1, 1, 5 ],
[ 4, 1, 1, 6 ], [ 4, 1, 1, 7 ], [ 4, 1, 1, 8 ], [ 4, 1, 1, 9 ],
[ 4, 1, 1, 10 ], [ 4, 1, 1, 11 ], [ 4, 1, 1, 12 ], [ 4, 1, 1, 13 ],
[ 4, 1, 2, 1 ], [ 4, 1, 2, 2 ], [ 4, 1, 2, 3 ], [ 4, 1, 2, 4 ],
[ 4, 1, 3, 1 ], [ 4, 1, 3, 2 ], [ 4, 1, 3, 3 ], [ 4, 1, 3, 4 ],
[ 4, 1, 3, 5 ], [ 4, 1, 3, 6 ], [ 4, 1, 3, 7 ], [ 4, 1, 3, 8 ],
[ 4, 1, 3, 9 ], [ 4, 1, 3, 10 ], [ 4, 1, 3, 11 ], [ 4, 1, 3, 12 ],
[ 4, 1, 4, 1 ], [ 4, 1, 4, 2 ], [ 4, 1, 4, 3 ], [ 4, 1, 4, 4 ],
[ 4, 1, 4, 5 ], [ 4, 1, 4, 6 ], [ 4, 1, 4, 7 ], [ 4, 1, 5, 1 ],
[ 4, 1, 5, 2 ], [ 4, 1, 5, 3 ], [ 4, 1, 6, 1 ], [ 4, 1, 6, 2 ],
[ 4, 1, 6, 3 ], [ 4, 1, 6, 4 ], [ 4, 1, 6, 5 ], [ 4, 2, 1, 1 ],
[ 4, 2, 1, 2 ], [ 4, 2, 1, 3 ], [ 4, 2, 1, 4 ], [ 4, 2, 1, 5 ],
[ 4, 2, 1, 6 ], [ 4, 2, 1, 7 ], [ 4, 2, 1, 8 ], [ 4, 2, 1, 9 ],
[ 4, 2, 1, 10 ], [ 4, 2, 1, 11 ], [ 4, 2, 1, 12 ], [ 4, 2, 1, 13 ],
[ 4, 2, 1, 14 ], [ 4, 2, 1, 15 ], [ 4, 2, 1, 16 ], [ 4, 2, 2, 1 ],
[ 4, 2, 2, 2 ], [ 4, 2, 2, 3 ], [ 4, 2, 2, 4 ], [ 4, 2, 3, 1 ],
[ 4, 2, 3, 2 ], [ 4, 2, 3, 3 ], [ 4, 2, 3, 4 ], [ 4, 2, 3, 5 ],
[ 4, 2, 3, 6 ], [ 4, 2, 3, 7 ], [ 4, 2, 3, 8 ], [ 4, 2, 4, 1 ],
[ 4, 2, 4, 2 ], [ 4, 2, 4, 3 ], [ 4, 2, 4, 4 ], [ 4, 2, 4, 5 ],
[ 4, 2, 4, 6 ], [ 4, 2, 5, 1 ], [ 4, 2, 5, 2 ], [ 4, 2, 5, 3 ],
[ 4, 2, 5, 4 ], [ 4, 2, 5, 5 ], [ 4, 2, 5, 6 ], [ 4, 2, 6, 1 ],
[ 4, 2, 6, 2 ], [ 4, 2, 7, 1 ], [ 4, 2, 7, 2 ], [ 4, 2, 7, 3 ],
[ 4, 3, 1, 1 ], [ 4, 3, 1, 2 ], [ 4, 3, 1, 3 ], [ 4, 3, 1, 4 ],
[ 4, 3, 1, 5 ], [ 4, 3, 1, 6 ], [ 4, 3, 2, 1 ], [ 4, 3, 2, 2 ],
[ 4, 3, 3, 1 ], [ 4, 3, 3, 2 ], [ 4, 3, 3, 3 ], [ 4, 3, 3, 4 ],
[ 4, 3, 4, 1 ], [ 4, 3, 4, 2 ], [ 4, 3, 4, 3 ], [ 4, 3, 5, 1 ],
[ 4, 3, 6, 1 ], [ 4, 3, 6, 2 ], [ 4, 4, 1, 1 ], [ 4, 4, 1, 2 ],
[ 4, 4, 1, 3 ], [ 4, 4, 1, 4 ], [ 4, 4, 1, 5 ], [ 4, 4, 1, 6 ],
[ 4, 4, 1, 7 ], [ 4, 4, 1, 8 ], [ 4, 4, 1, 9 ], [ 4, 4, 1, 10 ],
[ 4, 4, 1, 11 ], [ 4, 4, 1, 12 ], [ 4, 4, 1, 13 ], [ 4, 4, 1, 14 ],
[ 4, 4, 1, 15 ], [ 4, 4, 1, 16 ], [ 4, 4, 1, 17 ], [ 4, 4, 1, 18 ],
[ 4, 4, 1, 19 ], [ 4, 4, 1, 20 ], [ 4, 4, 1, 21 ], [ 4, 4, 1, 22 ],
[ 4, 4, 1, 23 ], [ 4, 4, 1, 24 ], [ 4, 4, 1, 25 ], [ 4, 4, 1, 26 ],
[ 4, 4, 1, 27 ], [ 4, 4, 1, 28 ], [ 4, 4, 1, 29 ], [ 4, 4, 1, 30 ],
[ 4, 4, 1, 31 ], [ 4, 4, 1, 32 ], [ 4, 4, 1, 33 ], [ 4, 4, 1, 34 ],
[ 4, 4, 1, 35 ], [ 4, 4, 1, 36 ], [ 4, 4, 1, 37 ], [ 4, 4, 1, 38 ],
[ 4, 4, 1, 39 ], [ 4, 4, 1, 40 ], [ 4, 4, 1, 41 ], [ 4, 4, 1, 42 ],
[ 4, 4, 1, 43 ], [ 4, 4, 1, 44 ], [ 4, 4, 1, 45 ], [ 4, 4, 1, 46 ],
[ 4, 4, 2, 1 ], [ 4, 4, 2, 2 ], [ 4, 4, 2, 3 ], [ 4, 4, 2, 4 ],
[ 4, 4, 2, 5 ], [ 4, 4, 2, 6 ], [ 4, 4, 2, 7 ], [ 4, 4, 2, 8 ],
[ 4, 4, 3, 1 ], [ 4, 4, 3, 2 ], [ 4, 4, 3, 3 ], [ 4, 4, 3, 4 ],
[ 4, 4, 3, 5 ], [ 4, 4, 3, 6 ], [ 4, 4, 3, 7 ], [ 4, 4, 3, 8 ],
[ 4, 4, 3, 9 ], [ 4, 4, 3, 10 ], [ 4, 4, 3, 11 ], [ 4, 4, 3, 12 ],
[ 4, 4, 3, 13 ], [ 4, 4, 3, 14 ], [ 4, 4, 3, 15 ], [ 4, 4, 3, 16 ],
[ 4, 4, 3, 17 ], [ 4, 4, 3, 18 ], [ 4, 4, 3, 19 ], [ 4, 4, 3, 20 ],
[ 4, 4, 3, 21 ], [ 4, 4, 3, 22 ], [ 4, 4, 3, 23 ], [ 4, 4, 3, 24 ],
[ 4, 4, 4, 1 ], [ 4, 4, 4, 2 ], [ 4, 4, 4, 3 ], [ 4, 4, 4, 4 ],
[ 4, 4, 4, 5 ], [ 4, 4, 4, 6 ], [ 4, 4, 4, 7 ], [ 4, 4, 4, 8 ],
[ 4, 4, 4, 9 ], [ 4, 4, 4, 10 ], [ 4, 4, 4, 11 ], [ 4, 4, 4, 12 ],
[ 4, 4, 4, 13 ], [ 4, 4, 4, 14 ], [ 4, 4, 5, 1 ], [ 4, 4, 5, 2 ],
[ 4, 4, 5, 3 ], [ 4, 4, 6, 1 ], [ 4, 4, 6, 2 ], [ 4, 4, 6, 3 ],
[ 4, 4, 6, 4 ], [ 4, 4, 6, 5 ], [ 5, 1, 1, 1 ], [ 5, 1, 2, 1 ],
[ 5, 1, 2, 2 ], [ 5, 1, 2, 3 ], [ 5, 1, 2, 4 ], [ 5, 1, 2, 5 ],
[ 5, 1, 2, 6 ], [ 5, 1, 2, 7 ], [ 5, 1, 2, 8 ], [ 5, 1, 2, 9 ],
[ 5, 1, 2, 10 ], [ 5, 1, 3, 1 ], [ 5, 1, 3, 2 ], [ 5, 1, 3, 3 ],
[ 5, 1, 3, 4 ], [ 5, 1, 3, 5 ], [ 5, 1, 3, 6 ], [ 5, 1, 4, 1 ],
[ 5, 1, 4, 2 ], [ 5, 1, 4, 3 ], [ 5, 1, 4, 4 ], [ 5, 1, 4, 5 ],
[ 5, 1, 4, 6 ], [ 5, 1, 5, 1 ], [ 5, 1, 5, 2 ], [ 5, 1, 5, 3 ],
[ 5, 1, 5, 4 ], [ 5, 1, 6, 1 ], [ 5, 1, 6, 2 ], [ 5, 1, 6, 3 ],
[ 5, 1, 6, 4 ], [ 5, 1, 6, 5 ], [ 5, 1, 6, 6 ], [ 5, 1, 7, 1 ],
[ 5, 1, 7, 2 ], [ 5, 1, 7, 3 ], [ 5, 1, 7, 4 ], [ 5, 1, 8, 1 ],
[ 5, 1, 8, 2 ], [ 5, 1, 9, 1 ], [ 5, 1, 9, 2 ], [ 5, 1, 10, 1 ],
[ 5, 1, 10, 2 ], [ 5, 1, 10, 3 ], [ 5, 1, 10, 4 ], [ 5, 1, 11, 1 ],
[ 5, 1, 11, 2 ], [ 5, 1, 12, 1 ], [ 5, 1, 12, 2 ], [ 5, 1, 13, 1 ],
[ 5, 1, 13, 2 ], [ 5, 2, 1, 1 ], [ 5, 2, 2, 1 ], [ 5, 2, 2, 2 ],
[ 5, 2, 2, 3 ], [ 5, 2, 2, 4 ], [ 5, 2, 2, 5 ], [ 5, 2, 2, 6 ],
[ 5, 2, 2, 7 ], [ 5, 2, 2, 8 ], [ 5, 2, 2, 9 ], [ 5, 2, 2, 10 ],
[ 5, 2, 2, 11 ], [ 5, 2, 2, 12 ], [ 5, 2, 2, 13 ], [ 5, 2, 2, 14 ],
[ 5, 2, 2, 15 ], [ 5, 2, 2, 16 ], [ 5, 2, 2, 17 ], [ 5, 2, 2, 18 ],
[ 5, 2, 2, 19 ], [ 5, 2, 3, 1 ], [ 5, 2, 3, 2 ], [ 5, 2, 3, 3 ],
[ 5, 2, 3, 4 ], [ 5, 2, 3, 5 ], [ 5, 2, 3, 6 ], [ 5, 2, 3, 7 ],
[ 5, 2, 3, 8 ], [ 5, 2, 3, 9 ], [ 5, 2, 3, 10 ], [ 5, 2, 3, 11 ],
[ 5, 2, 3, 12 ], [ 5, 2, 3, 13 ], [ 5, 2, 3, 14 ], [ 5, 2, 4, 1 ],
[ 5, 2, 4, 2 ], [ 5, 2, 4, 3 ], [ 5, 2, 4, 4 ], [ 5, 2, 4, 5 ],
[ 5, 2, 4, 6 ], [ 5, 2, 4, 7 ], [ 5, 2, 4, 8 ], [ 5, 2, 4, 9 ],
[ 5, 2, 4, 10 ], [ 5, 2, 5, 1 ], [ 5, 2, 5, 2 ], [ 5, 2, 5, 3 ],
[ 5, 2, 5, 4 ], [ 5, 2, 5, 5 ], [ 5, 2, 5, 6 ], [ 5, 2, 6, 1 ],
[ 5, 2, 6, 2 ], [ 5, 2, 6, 3 ], [ 5, 2, 7, 1 ], [ 5, 2, 7, 2 ],
[ 5, 2, 7, 3 ], [ 5, 2, 7, 4 ], [ 5, 2, 8, 1 ], [ 5, 2, 8, 2 ],
[ 5, 2, 8, 3 ], [ 5, 2, 9, 1 ], [ 5, 2, 9, 2 ], [ 5, 2, 9, 3 ],
[ 6, 1, 1, 1 ], [ 6, 1, 1, 2 ], [ 6, 1, 1, 3 ], [ 6, 1, 1, 4 ],
[ 6, 1, 1, 5 ], [ 6, 1, 1, 6 ], [ 6, 1, 1, 7 ], [ 6, 1, 1, 8 ],
[ 6, 1, 1, 9 ], [ 6, 1, 1, 10 ], [ 6, 1, 1, 11 ], [ 6, 1, 1, 12 ],
[ 6, 1, 1, 13 ], [ 6, 1, 1, 14 ], [ 6, 1, 1, 15 ], [ 6, 1, 1, 16 ],
[ 6, 1, 1, 17 ], [ 6, 1, 1, 18 ], [ 6, 1, 1, 19 ], [ 6, 1, 1, 20 ],
[ 6, 1, 1, 21 ], [ 6, 1, 1, 22 ], [ 6, 1, 1, 23 ], [ 6, 1, 1, 24 ],
[ 6, 1, 1, 25 ], [ 6, 1, 1, 26 ], [ 6, 1, 1, 27 ], [ 6, 1, 1, 28 ],
[ 6, 1, 1, 29 ], [ 6, 1, 1, 30 ], [ 6, 1, 1, 31 ], [ 6, 1, 1, 32 ],
[ 6, 1, 1, 33 ], [ 6, 1, 1, 34 ], [ 6, 1, 1, 35 ], [ 6, 1, 1, 36 ],
[ 6, 1, 1, 37 ], [ 6, 1, 1, 38 ], [ 6, 1, 1, 39 ], [ 6, 1, 1, 40 ],
[ 6, 1, 1, 41 ], [ 6, 1, 1, 42 ], [ 6, 1, 1, 43 ], [ 6, 1, 1, 44 ],
[ 6, 1, 1, 45 ], [ 6, 1, 1, 46 ], [ 6, 1, 1, 47 ], [ 6, 1, 1, 48 ],
[ 6, 1, 1, 49 ], [ 6, 1, 1, 50 ], [ 6, 1, 1, 51 ], [ 6, 1, 1, 52 ],
[ 6, 1, 1, 53 ], [ 6, 1, 1, 54 ], [ 6, 1, 1, 55 ], [ 6, 1, 1, 56 ],
[ 6, 1, 1, 57 ], [ 6, 1, 1, 58 ], [ 6, 1, 1, 59 ], [ 6, 1, 1, 60 ],
[ 6, 1, 1, 61 ], [ 6, 1, 1, 62 ], [ 6, 1, 1, 63 ], [ 6, 1, 1, 64 ],
[ 6, 1, 1, 65 ], [ 6, 1, 1, 66 ], [ 6, 1, 1, 67 ], [ 6, 1, 1, 68 ],
[ 6, 1, 1, 69 ], [ 6, 1, 1, 70 ], [ 6, 1, 1, 71 ], [ 6, 1, 1, 72 ],
[ 6, 1, 1, 73 ], [ 6, 1, 1, 74 ], [ 6, 1, 1, 75 ], [ 6, 1, 1, 76 ],
[ 6, 1, 1, 77 ], [ 6, 1, 1, 78 ], [ 6, 1, 1, 79 ], [ 6, 1, 1, 80 ],
[ 6, 1, 1, 81 ], [ 6, 1, 1, 82 ], [ 6, 1, 1, 83 ], [ 6, 1, 1, 84 ],
[ 6, 1, 1, 85 ], [ 6, 1, 1, 86 ], [ 6, 1, 1, 87 ], [ 6, 1, 1, 88 ],
[ 6, 1, 1, 89 ], [ 6, 1, 1, 90 ], [ 6, 1, 1, 91 ], [ 6, 1, 1, 92 ],
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[ 32, 14, 4, 3 ], [ 32, 14, 4, 4 ], [ 32, 14, 4, 5 ], [ 32, 14, 4, 6 ],
[ 32, 14, 4, 7 ], [ 32, 14, 4, 8 ], [ 32, 14, 4, 9 ], [ 32, 14, 4, 10 ],
[ 32, 15, 1, 1 ], [ 32, 15, 1, 2 ], [ 32, 15, 1, 3 ], [ 32, 15, 1, 4 ],
[ 32, 15, 2, 1 ], [ 32, 15, 2, 2 ], [ 32, 15, 2, 3 ], [ 32, 15, 2, 4 ],
[ 32, 16, 1, 1 ], [ 32, 16, 2, 1 ], [ 32, 16, 2, 2 ], [ 32, 16, 3, 1 ],
[ 32, 17, 1, 1 ], [ 32, 17, 1, 2 ], [ 32, 17, 1, 3 ], [ 32, 17, 1, 4 ],
[ 32, 17, 1, 5 ], [ 32, 17, 1, 6 ], [ 32, 17, 1, 7 ], [ 32, 17, 1, 8 ],
[ 32, 17, 2, 1 ], [ 32, 17, 2, 2 ], [ 32, 17, 2, 3 ], [ 32, 17, 2, 4 ],
[ 32, 17, 2, 5 ], [ 32, 17, 2, 6 ], [ 32, 17, 2, 7 ], [ 32, 17, 2, 8 ],
[ 32, 18, 1, 1 ], [ 32, 18, 1, 2 ], [ 32, 18, 2, 1 ], [ 32, 18, 3, 1 ],
[ 32, 18, 3, 2 ], [ 32, 19, 1, 1 ], [ 32, 19, 1, 2 ], [ 32, 19, 2, 1 ],
[ 32, 19, 2, 2 ], [ 32, 19, 3, 1 ], [ 32, 19, 3, 2 ], [ 32, 20, 1, 1 ],
[ 32, 20, 1, 2 ], [ 32, 20, 2, 1 ], [ 32, 20, 2, 2 ], [ 32, 20, 3, 1 ],
[ 32, 21, 1, 1 ], [ 32, 21, 1, 2 ], [ 32, 21, 1, 3 ], [ 32, 21, 1, 4 ],
[ 32, 21, 2, 1 ], [ 32, 21, 2, 2 ], [ 32, 21, 3, 1 ], [ 32, 21, 3, 2 ],
[ 33, 1, 1, 1 ], [ 33, 2, 1, 1 ], [ 33, 3, 1, 1 ], [ 33, 3, 1, 2 ],
[ 33, 4, 1, 1 ], [ 33, 5, 1, 1 ], [ 33, 5, 1, 2 ], [ 33, 5, 1, 3 ],
[ 33, 6, 1, 1 ], [ 33, 6, 1, 2 ], [ 33, 7, 1, 1 ], [ 33, 8, 1, 1 ],
[ 33, 8, 1, 2 ], [ 33, 9, 1, 1 ], [ 33, 9, 1, 2 ], [ 33, 10, 1, 1 ],
[ 33, 10, 1, 2 ], [ 33, 11, 1, 1 ], [ 33, 12, 1, 1 ], [ 33, 12, 1, 2 ],
[ 33, 13, 1, 1 ], [ 33, 14, 1, 1 ], [ 33, 14, 2, 1 ], [ 33, 15, 1, 1 ],
[ 33, 16, 1, 1 ] ];
cryst3params :=
[ [ 1, 1, 1, 1 ], [ 1, 2, 1, 1 ], [ 2, 1, 1, 1 ], [ 2, 1, 1, 2 ],
[ 2, 1, 2, 1 ], [ 2, 2, 1, 1 ], [ 2, 2, 1, 2 ], [ 2, 2, 2, 1 ],
[ 2, 2, 2, 2 ], [ 2, 3, 1, 1 ], [ 2, 3, 1, 2 ], [ 2, 3, 1, 3 ],
[ 2, 3, 1, 4 ], [ 2, 3, 2, 1 ], [ 2, 3, 2, 2 ], [ 3, 1, 1, 1 ],
[ 3, 1, 1, 2 ], [ 3, 1, 1, 3 ], [ 3, 1, 1, 4 ], [ 3, 1, 2, 1 ],
[ 3, 1, 2, 2 ], [ 3, 1, 3, 1 ], [ 3, 1, 4, 1 ], [ 3, 1, 4, 2 ],
[ 3, 2, 1, 1 ], [ 3, 2, 1, 2 ], [ 3, 2, 1, 3 ], [ 3, 2, 1, 4 ],
[ 3, 2, 1, 5 ], [ 3, 2, 1, 6 ], [ 3, 2, 1, 7 ], [ 3, 2, 1, 8 ],
[ 3, 2, 1, 9 ], [ 3, 2, 1, 10 ], [ 3, 2, 2, 1 ], [ 3, 2, 2, 2 ],
[ 3, 2, 2, 3 ], [ 3, 2, 3, 1 ], [ 3, 2, 3, 2 ], [ 3, 2, 3, 3 ],
[ 3, 2, 3, 4 ], [ 3, 2, 4, 1 ], [ 3, 2, 4, 2 ], [ 3, 2, 5, 1 ],
[ 3, 2, 5, 2 ], [ 3, 2, 5, 3 ], [ 3, 3, 1, 1 ], [ 3, 3, 1, 2 ],
[ 3, 3, 1, 3 ], [ 3, 3, 1, 4 ], [ 3, 3, 1, 5 ], [ 3, 3, 1, 6 ],
[ 3, 3, 1, 7 ], [ 3, 3, 1, 8 ], [ 3, 3, 1, 9 ], [ 3, 3, 1, 10 ],
[ 3, 3, 1, 11 ], [ 3, 3, 1, 12 ], [ 3, 3, 1, 13 ], [ 3, 3, 1, 14 ],
[ 3, 3, 1, 15 ], [ 3, 3, 1, 16 ], [ 3, 3, 2, 1 ], [ 3, 3, 2, 2 ],
[ 3, 3, 2, 3 ], [ 3, 3, 2, 4 ], [ 3, 3, 2, 5 ], [ 3, 3, 2, 6 ],
[ 3, 3, 3, 1 ], [ 3, 3, 3, 2 ], [ 3, 3, 4, 1 ], [ 3, 3, 4, 2 ],
[ 3, 3, 4, 3 ], [ 3, 3, 4, 4 ], [ 4, 1, 1, 1 ], [ 4, 1, 1, 2 ],
[ 4, 1, 1, 3 ], [ 4, 1, 2, 1 ], [ 4, 1, 2, 2 ], [ 4, 2, 1, 1 ],
[ 4, 2, 2, 1 ], [ 4, 3, 1, 1 ], [ 4, 3, 1, 2 ], [ 4, 3, 1, 3 ],
[ 4, 3, 1, 4 ], [ 4, 3, 2, 1 ], [ 4, 3, 2, 2 ], [ 4, 4, 1, 1 ],
[ 4, 4, 1, 2 ], [ 4, 4, 1, 3 ], [ 4, 4, 1, 4 ], [ 4, 4, 1, 5 ],
[ 4, 4, 1, 6 ], [ 4, 4, 2, 1 ], [ 4, 4, 2, 2 ], [ 4, 5, 1, 1 ],
[ 4, 5, 1, 2 ], [ 4, 5, 1, 3 ], [ 4, 5, 1, 4 ], [ 4, 5, 1, 5 ],
[ 4, 5, 1, 6 ], [ 4, 5, 1, 7 ], [ 4, 5, 1, 8 ], [ 4, 5, 2, 1 ],
[ 4, 5, 2, 2 ], [ 4, 5, 2, 3 ], [ 4, 5, 2, 4 ], [ 4, 6, 1, 1 ],
[ 4, 6, 1, 2 ], [ 4, 6, 1, 3 ], [ 4, 6, 1, 4 ], [ 4, 6, 2, 1 ],
[ 4, 6, 2, 2 ], [ 4, 6, 2, 3 ], [ 4, 6, 2, 4 ], [ 4, 6, 3, 1 ],
[ 4, 6, 3, 2 ], [ 4, 6, 4, 1 ], [ 4, 6, 4, 2 ], [ 4, 7, 1, 1 ],
[ 4, 7, 1, 2 ], [ 4, 7, 1, 3 ], [ 4, 7, 1, 4 ], [ 4, 7, 1, 5 ],
[ 4, 7, 1, 6 ], [ 4, 7, 1, 7 ], [ 4, 7, 1, 8 ], [ 4, 7, 1, 9 ],
[ 4, 7, 1, 10 ], [ 4, 7, 1, 11 ], [ 4, 7, 1, 12 ], [ 4, 7, 1, 13 ],
[ 4, 7, 1, 14 ], [ 4, 7, 1, 15 ], [ 4, 7, 1, 16 ], [ 4, 7, 2, 1 ],
[ 4, 7, 2, 2 ], [ 4, 7, 2, 3 ], [ 4, 7, 2, 4 ], [ 5, 1, 1, 1 ],
[ 5, 1, 2, 1 ], [ 5, 1, 2, 2 ], [ 5, 2, 1, 1 ], [ 5, 2, 2, 1 ],
[ 5, 3, 1, 1 ], [ 5, 3, 2, 1 ], [ 5, 3, 2, 2 ], [ 5, 3, 3, 1 ],
[ 5, 3, 3, 2 ], [ 5, 4, 1, 1 ], [ 5, 4, 1, 2 ], [ 5, 4, 2, 1 ],
[ 5, 4, 2, 2 ], [ 5, 4, 3, 1 ], [ 5, 4, 3, 2 ], [ 5, 5, 1, 1 ],
[ 5, 5, 1, 2 ], [ 5, 5, 2, 1 ], [ 5, 5, 2, 2 ], [ 5, 5, 3, 1 ],
[ 5, 5, 3, 2 ], [ 6, 1, 1, 1 ], [ 6, 1, 1, 2 ], [ 6, 1, 1, 3 ],
[ 6, 1, 1, 4 ], [ 6, 2, 1, 1 ], [ 6, 3, 1, 1 ], [ 6, 3, 1, 2 ],
[ 6, 4, 1, 1 ], [ 6, 4, 1, 2 ], [ 6, 4, 1, 3 ], [ 6, 4, 1, 4 ],
[ 6, 5, 1, 1 ], [ 6, 5, 1, 2 ], [ 6, 5, 1, 3 ], [ 6, 5, 1, 4 ],
[ 6, 6, 1, 1 ], [ 6, 6, 1, 2 ], [ 6, 6, 2, 1 ], [ 6, 6, 2, 2 ],
[ 6, 7, 1, 1 ], [ 6, 7, 1, 2 ], [ 6, 7, 1, 3 ], [ 6, 7, 1, 4 ],
[ 7, 1, 1, 1 ], [ 7, 1, 1, 2 ], [ 7, 1, 2, 1 ], [ 7, 1, 3, 1 ],
[ 7, 1, 3, 2 ], [ 7, 2, 1, 1 ], [ 7, 2, 1, 2 ], [ 7, 2, 1, 3 ],
[ 7, 2, 2, 1 ], [ 7, 2, 2, 2 ], [ 7, 2, 3, 1 ], [ 7, 2, 3, 2 ],
[ 7, 3, 1, 1 ], [ 7, 3, 1, 2 ], [ 7, 3, 1, 3 ], [ 7, 3, 2, 1 ],
[ 7, 3, 2, 2 ], [ 7, 3, 3, 1 ], [ 7, 3, 3, 2 ], [ 7, 4, 1, 1 ],
[ 7, 4, 1, 2 ], [ 7, 4, 2, 1 ], [ 7, 4, 2, 2 ], [ 7, 4, 3, 1 ],
[ 7, 4, 3, 2 ], [ 7, 5, 1, 1 ], [ 7, 5, 1, 2 ], [ 7, 5, 1, 3 ],
[ 7, 5, 1, 4 ], [ 7, 5, 2, 1 ], [ 7, 5, 2, 2 ], [ 7, 5, 2, 3 ],
[ 7, 5, 2, 4 ], [ 7, 5, 3, 1 ], [ 7, 5, 3, 2 ] ];
cryst2params :=
[ [ 1, 1, 1, 1 ], [ 1, 2, 1, 1 ], [ 2, 1, 1, 1 ], [ 2, 1, 1, 2 ],
[ 2, 1, 2, 1 ], [ 2, 2, 1, 1 ], [ 2, 2, 1, 2 ], [ 2, 2, 1, 3 ],
[ 2, 2, 2, 1 ], [ 3, 1, 1, 1 ], [ 3, 2, 1, 1 ], [ 3, 2, 1, 2 ],
[ 4, 1, 1, 1 ], [ 4, 2, 1, 1 ], [ 4, 2, 2, 1 ], [ 4, 3, 1, 1 ],
[ 4, 4, 1, 1 ] ];
NumberSpaceGroups := function( dim )
if dim = 4 then return 4783; fi;
if dim = 3 then return 219; fi;
if dim = 2 then return 17; fi;
end;
SpaceGroup := function( dim, nr )
local type;
if dim = 4 then
type := cryst4params[nr];
return SpaceGroupBBNWZ( 4, type[1], type[2], type[3], type[4] );
elif dim = 3 then
type := cryst3params[nr];
return SpaceGroupBBNWZ( 3, type[1], type[2], type[3], type[4] );
elif dim = 2 then
type := cryst2params[nr];
return SpaceGroupBBNWZ( 2, type[1], type[2], type[3], type[4] );
else
return fail;
fi;
end;
PointGroupByNumber := function( dim, nr )
local type;
if dim = 4 then
type := cryst4params[nr];
return MatGroupZClass( 4, type[1], type[2], type[3] );
elif dim = 3 then
type := cryst3params[nr];
return MatGroupZClass( 3, type[1], type[2], type[3] );
elif dim = 2 then
type := cryst2params[nr];
return MatGroupZClass( 2, type[1], type[2], type[3] );
fi;
end;
SpaceGroupPcpGroup := function( dim, nr )
local G, iso, H, F, n;
G := SpaceGroup( dim, nr );
iso := IsomorphismPcpGroup( G );
if IsBool( iso ) then return fail; fi;
H := Image( iso );
n := Length( Igs(H) );
F := Subgroup( H, Igs(H){[n-dim+1..n]} );
SetFittingSubgroup( H, F );
return H;
end;
aclib-1.3.2/gap/groups.gd 0000664 0003717 0003717 00000002125 13627151544 016413 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W groups.gd Karel Dekimpe
#W Bettina Eick
##
DeclareProperty( "IsAlmostCrystallographic", IsGroup );
DeclareProperty( "IsAlmostBieberbachGroup", IsGroup );
DeclareAttribute( "AlmostCrystallographicInfo", IsGroup );
DeclareAttribute( "NaturalHomomorphismOnHolonomyGroup", IsGroup );
DeclareAttribute( "HolonomyGroup", IsGroup );
DeclareGlobalFunction( "AlmostCrystallographicDim3" );
DeclareGlobalFunction( "AlmostCrystallographicDim4" );
DeclareGlobalFunction( "AlmostCrystallographicGroup" );
DeclareGlobalFunction( "AlmostCrystallographicPcpDim3" );
DeclareGlobalFunction( "AlmostCrystallographicPcpDim4" );
DeclareGlobalFunction( "AlmostCrystallographicPcpGroup" );
DeclareGlobalFunction( "IsolatorSubgroup" );
DeclareAttribute( "OrientationModule", IsGroup );
DeclareOperation( "BettiNumber", [IsGroup, IsInt] );
DeclareAttribute( "BettiNumbers", IsGroup );
DeclareGlobalFunction( "HasExtensionOfType" );
aclib-1.3.2/gap/union.gi 0000664 0003717 0003717 00000011357 13627151544 016240 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W union.gi Karel Dekimpe
#W Bettina Eick
##
#############################################################################
##
#F GenericDeterminantMat( mat )
##
GenericDeterminantMat := function( mat )
local d, det, sig, i, sub;
# set up
d := Length( mat );
if ForAny( mat, x -> Length(x) <> d ) then return fail; fi;
# the trivial cases
if d = 1 then return mat[1][1]; fi;
if d = 2 then return mat[1][1] * mat[2][2] - mat[1][2] * mat[2][1]; fi;
# otherwise use first row and recursion
det := 0;
sig := 1;
for i in [1..d] do
sub := Concatenation( [1..i-1], [i+1..d] );
sub := mat{[2..d]}{sub};
det := det + sig * mat[1][i] * GenericDeterminantMat( sub );
sig := - sig;
od;
return det;
end;
#############################################################################
##
#F NullspaceIntMod( base, vec, p ) . . . . . . . . . . . . b * vec = 0 mod p
##
## computes those elements b in with b * vec = 0 mod p. Returns a
## triangulized basis with respect to .
##
NullspaceIntMod := function( base, vec, p )
local imgs, d, null;
# get images
imgs := List( base, x -> x * vec );
imgs := List( imgs, x -> x mod p );
d := Length( imgs );
if ForAll( imgs, x -> x = 0 ) then return IdentityMat( d ); fi;
# compute kernel of imgs vector - must have rank d-1
Add( imgs, p );
null := NullspaceIntMat( TransposedMat( [imgs] ) );
null := List( null, x -> x{[1..d]} );
null := NormalFormIntMat( null, 0 ).normal;
# return this images nullspace
return Filtered( null, x -> PositionNonZero(x) <= d );
end;
#############################################################################
##
#F FindMaximals( sub ) . . . . . . . .subgroups which are maximal within sub
##
FindMaximals := function( sub )
local new, i, tmp;
new := [];
for i in [1..Length(sub)] do
if Size( sub[i] ) > 1 then
if Length(new) = 0 then
Add( new, sub[i] );
elif not ForAny( new, x -> IsSubset( x, sub[i] ) ) then
tmp := Filtered( new, x -> not IsSubset( sub[i], x ) );
Add( new, sub[i] );
fi;
fi;
od;
return new;
end;
if not IsBound( SizeOfUnion ) then SizeOfUnion := false; fi;
#############################################################################
##
#F SizeOfUnionRec( sub ) -- recursive version
##
SizeOfUnionRec := function( list )
local s, i, int, t, n;
if Length( list ) = 0 then return 1; fi;
s := Size( list[1] );
n := Length( list );
for i in [2..n] do
int := List( list{[1..i-1]}, x -> Intersection( x, list[i]));
t := SizeOfUnion( int );
s := s + Size( list[i] ) - t;
od;
return s;
end;
#############################################################################
##
#F SizeOfUnionTriv -- trivial version
##
SizeOfUnionTriv := function( list )
return Length( Union( List( list, Elements ) ) );
end;
#############################################################################
##
#F SizeOfUnion -- main function
##
SizeOfUnion := function( sub )
local list;
list := FindMaximals( sub );
if Length(list) = 0 then return 1; fi;
if Length(list) = 1 then return Size(list[1]); fi;
if Sum(List(list, Size)) < 2000 then
return SizeOfUnionTriv(list);
fi;
return SizeOfUnionRec(list);
end;
#############################################################################
##
#F SizeOfUnionMod( subs, e ) . . . . . . . . .size of the union of subs mod e
##
## is a list of bases containing (eZ)^d. Compute the size of the union
## of in (Z/eZ)^d.
##
## This function needs to be profiled.
##
SizeOfUnionMod := function( subs, e )
local d, F, V, b, news;
# the trivial case
if Length( subs ) = 0 then return 1; fi;
d := Length( subs[1] );
if IsPrimeInt( e ) then
F := GF(e);
V := F^d;
b := BasisVectors( Basis( V ) );
news := List( subs, x -> x * b );
news := List( news, x -> Subspace( V, x ) );
if ForAny( news, x -> Size(x) = e^d ) then return e^d; fi;
# return SizeOfUnion( news );
return Length( Union( List( news, Elements ) ) );
fi;
V := AbelianGroup( List( [1..d], x -> e ) );
b := GeneratorsOfGroup(V);
news := List( subs, x -> List( x, y -> MappedVector( y, b ) ) );
news := List( news, x -> Subgroup( V, x ) );
if ForAny( news, x -> Size(x) = e^d ) then return e^d; fi;
# return SizeOfUnion( news );
return Length( Union( List( news, Elements ) ) );
end;
aclib-1.3.2/gap/pcpgrp4.gi 0000664 0003717 0003717 00000246504 13627151544 016473 0 ustar gap-jenkins gap-jenkins #############################################################################
##
#W pcpgrp4.gi Karel Dekimpe
#W Bettina Eick
##
## This file contains the 4-dimensional almost crystallographic groups
## as pcp groups. There are 95 types of groups.
##
ACPcpGroupDim4Nr001 := function (k1, k2, k3)
local FTL;
FTL := FromTheLeftCollector( 4 );
SetConjugate( FTL, 2, 1, [2,1, 4,k1 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,k2 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,k3 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr002 := function (k1, k2, k3 , k4, k5, k6, k7)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k7 ] );
SetConjugate( FTL, 2, 1, [2,-1, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,0, 5,k5 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k6 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k2 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,k3 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr003 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr004 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,1, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr004b:= function (k1, k2, k3 )
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,1, 4,0, 5,k3 ] );
SetConjugate( FTL, 2, 1, [2,-1, 3,0, 4,0, 5,k1 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,1, 4,0, 5,2*k3 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,-k2 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,-1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,2*k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,2*k2 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr005 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,-1, 4,0, 5,k2 ] );
SetConjugate( FTL, 3, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr006 := function (k1, k2, k3 )
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k3 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,0, 5,0 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr007 := function (k1, k2, k3 )
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,1, 5,k3 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,0, 5,-k1 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,2*k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr007b:= function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,1, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,0, 5,-k3 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,2*k4 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,-1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,2*k2 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr008 := function (k1, k2, k3)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k3 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,1, 4,0, 5,k2 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr009 := function (k1, k2, k3)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,1, 5,k3 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,1, 4,0, 5,-k1 + k2 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr009b:= function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,1, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,1, 4,0, 5,k2 - k3 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k3 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,2*k4 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,-1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k2 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,-k2 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr010 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,k5 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr011 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,-1, 5,0, 6,-k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-2*k6 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,1, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr012 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,2*k2 - k5 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr013 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,1, 6,k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k1 + k2 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,k4 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,-2*k6 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,-2*k6 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr014 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,-1, 5,1, 6,-k3 - k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k1 + k2 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k3 - 2*k6 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,1, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr014b:= function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,-1, 5,1, 6,-k2 - 2*k3 + 2*k5 - k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,k2 - 2*k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k2 + 2*k3 - 2*k5 + 2*k6 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,1, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,1, 5,0, 6,2*k3 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,-k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,2*k2 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr015 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,1, 6,k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,k1 + 2*k2 - k4 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,-2*k6 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,-2*k6 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr018 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,1, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,0, 6,k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,1, 5,0, 6,2*k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,0 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,-k1 + 2*k2 - 2*k3 + 2*k4 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,k1 - 2*k3 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr019 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,1, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,-1, 6,-3*k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,3*k1 + 2*k2 - 2*k3 + 2*k4 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,-k1 - 2*k2 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr019b:= function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,1, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,-1, 6,k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,1, 5,0, 6,2*k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,0 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,-k1 + 2*k2 - 2*k3 + 2*k4 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,k1 - 2*k3 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr019c:= function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,1, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,-1, 6,-2*k2 + 2*k3 + k4 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,1, 5,0, 6,2*k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,-k1 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,2*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr026 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr027 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,-k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,k3 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr029 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,1, 5,0, 6,-2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,-k1 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,2*(k2 - k3 + k4) ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr029b:= function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,1, 5,0, 6,-2*k3 + 2*k4 - k5 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,-k1 - k2 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k4 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,2*(k3 - k4 + k5) ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr029c:= function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k1 + k2 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,1, 5,0, 6,-4*k1 + k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,-k3 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*(k1 + k2) ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,4*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr030 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,1, 5,0, 6,-2*k3 - k5 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,-k1 - k2 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k4 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,2*(k3 + k5) ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr031 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,1, 5,0, 6,2*k3 - k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-2*(k3 - k4) ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr032 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,0, 6,3*k1 - 2*k2 + 2*k4 - k5 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,2*k4 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k1 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,-k3 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,-k1 - 2*k4 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,-3*k1 + 2*k2 - 2*k4 + 2*k5 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr033 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,0, 6,2*k3 - k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k1 - 2*k3 + 2*k4 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,k1 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr033b:= function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,0, 6,3*k1 - 2*k2 - k3 + 2*k4 - k5 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,2*k4 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k1 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,-k3 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,-k1 - 2*k4 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,-3*k1 + 2*k2 + k3 - 2*k4 + 2*k5 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr033c:= function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,0, 6,-k1 + 2*k3 + k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,2*k3 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,k1 + 2*k2 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,2*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr034 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,1, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,0, 6,2*k1 - k2 - 2*k3 - k5 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,-k1 - k2 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k1 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,k1 + k2 + 2*k4 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,-2*k1 + k2 + 2*k3 + 2*k5 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr036 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,1, 5,0, 6,k3 ] );
SetConjugate( FTL, 4, 1, [3,1, 4,0, 5,0, 6,-k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,2*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr037 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,-k3 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,1, 5,0, 6,-k4 ] );
SetConjugate( FTL, 4, 1, [3,1, 4,0, 5,0, 6,-k4 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,-k2 + 2*k4 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr041 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,1, 5,1, 6,k1 - k2 - k3 + k5 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,-1, 5,0, 6,2*k1 - k4 ] );
SetConjugate( FTL, 4, 1, [3,-1, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,2*(k1 - k5) ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,2*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr043 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,1, 5,0, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,1, 6,-k2 - 2*k3 - k5 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,1, 5,-1, 6,k1 + k2 + 2*k4 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,1, 5,0, 6,2*k4 ] );
SetConjugate( FTL, 5, 1, [3,-1, 4,1, 5,0, 6,-k2 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,1, 5,-1, 6,k2 + 2*k3 + 2*k5 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,-1, 6,k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,2*(k2 + k3 + k5) ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,-k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr045 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,1, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,-k3 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,0, 5,-1, 6,-k4 ] );
SetConjugate( FTL, 4, 1, [3,1, 4,1, 5,1, 6,k4 + 2*k5 ] );
SetConjugate( FTL, 5, 1, [3,-1, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,1, 5,0, 6,k2 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,-1, 4,-1, 5,-1, 6,k2 - 2*k4 - 2*k5 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,-k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr055 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 7 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [4,0, 5,0, 6,0, 7,2*k1 + k2 + k4 + k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 4,1, 5,-1, 6,0, 7,k1 + 2*k2 - 2*k3 + 2*k4 + k6 ] );
SetConjugate( FTL, 3, 1, [3, 1 , 4,0, 5,0, 6,0, 7,0 ] );
SetConjugate( FTL, 4, 1, [4,-1, 5,0, 6,0, 7,-k1 - 2*k2 + 2*k3 - 2*k4 ] );
SetConjugate( FTL, 5, 1, [4,0, 5,-1, 6,0, 7,-k1 - 2*k3 ] );
SetConjugate( FTL, 6, 1, [4,0, 5,0, 6,-1, 7,k5 ] );
SetConjugate( FTL, 7, 1, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [4,0, 5,1, 6,0, 7,k3 ] );
SetConjugate( FTL, 3, 2, [ 3, 1, 4,-1, 5,1, 6,0, 7,-k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 4, 2, [4,-1, 5,0, 6,0, 7,k1 ] );
SetConjugate( FTL, 5, 2, [4,0, 5,1, 6,0, 7,2*k3 ] );
SetConjugate( FTL, 6, 2, [4,0, 5,0, 6,-1, 7,0 ] );
SetConjugate( FTL, 7, 2, [4,0, 5,0, 6,0, 7,-1 ] );
SetRelativeOrder( FTL, 3, 2 );
SetPower( FTL, 3, [4,0, 5,0, 6,0, 7,k2 ] );
SetConjugate( FTL, 4, 3, [4,-1, 5,0, 6,0, 7,-k1 - 2*k2 + 2*k3 - 2*k4 ] );
SetConjugate( FTL, 5, 3, [4,0, 5,-1, 6,0, 7,-k1 - 2*k3 ] );
SetConjugate( FTL, 6, 3, [4,0, 5,0, 6,1, 7,0 ] );
SetConjugate( FTL, 7, 3, [4,0, 5,0, 6,0, 7,1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0, 7,2*k1 ] );
SetConjugate( FTL, 6, 4, [6,1, 7,0 ] );
SetConjugate( FTL, 6, 5, [6,1, 7,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr056 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 7 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [4,0, 5,0, 6,0, 7,k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 4,0, 5,-1, 6,1, 7,-2*k3 - k5 + 2*k6 ] );
SetConjugate( FTL, 3, 1, [3, 1 , 4,1, 5,1, 6,0, 7,k4 ] );
SetConjugate( FTL, 4, 1, [4,-1, 5,0, 6,0, 7,2*(k1 + k3 - k4) ] );
SetConjugate( FTL, 5, 1, [4,0, 5,-1, 6,0, 7,-2*k3 ] );
SetConjugate( FTL, 6, 1, [4,0, 5,0, 6,-1, 7,2*(k3 + k5 - k6) ] );
SetConjugate( FTL, 7, 1, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [4,0, 5,1, 6,0, 7,k3 ] );
SetConjugate( FTL, 3, 2, [ 3, 1, 4,1, 5,1, 6,0, 7,-k2 + k4 ] );
SetConjugate( FTL, 4, 2, [4,-1, 5,0, 6,0, 7,k1 ] );
SetConjugate( FTL, 5, 2, [4,0, 5,1, 6,0, 7,2*k3 ] );
SetConjugate( FTL, 6, 2, [4,0, 5,0, 6,-1, 7,0 ] );
SetConjugate( FTL, 7, 2, [4,0, 5,0, 6,0, 7,-1 ] );
SetRelativeOrder( FTL, 3, 2 );
SetPower( FTL, 3, [4,0, 5,0, 6,0, 7,k2 ] );
SetConjugate( FTL, 4, 3, [4,-1, 5,0, 6,0, 7,k1 + 2*k3 - 2*k4 ] );
SetConjugate( FTL, 5, 3, [4,0, 5,-1, 6,0, 7,k1 - 2*k3 ] );
SetConjugate( FTL, 6, 3, [4,0, 5,0, 6,1, 7,0 ] );
SetConjugate( FTL, 7, 3, [4,0, 5,0, 6,0, 7,1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0, 7,2*k1 ] );
SetConjugate( FTL, 6, 4, [6,1, 7,0 ] );
SetConjugate( FTL, 6, 5, [6,1, 7,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr058 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 7 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [4,0, 5,0, 6,0, 7,k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 4,1, 5,-1, 6,1, 7,-3*k1 - 2*k3 - k5 + 2*k6 ] );
SetConjugate( FTL, 3, 1, [3, 1 , 4,0, 5,0, 6,0, 7,0 ] );
SetConjugate( FTL, 4, 1, [4,-1, 5,0, 6,0, 7,-k1 - 2*k2 + 2*k3 - 2*k4 ] );
SetConjugate( FTL, 5, 1, [4,0, 5,-1, 6,0, 7,-k1 - 2*k3 ] );
SetConjugate( FTL, 6, 1, [4,0, 5,0, 6,-1, 7,2*(2*k1 + k2 + k4 + k5 - k6) ] );
SetConjugate( FTL, 7, 1, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [4,0, 5,1, 6,0, 7,k3 ] );
SetConjugate( FTL, 3, 2, [ 3, 1, 4,-1, 5,1, 6,0, 7,-k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 4, 2, [4,-1, 5,0, 6,0, 7,k1 ] );
SetConjugate( FTL, 5, 2, [4,0, 5,1, 6,0, 7,2*k3 ] );
SetConjugate( FTL, 6, 2, [4,0, 5,0, 6,-1, 7,0 ] );
SetConjugate( FTL, 7, 2, [4,0, 5,0, 6,0, 7,-1 ] );
SetRelativeOrder( FTL, 3, 2 );
SetPower( FTL, 3, [4,0, 5,0, 6,0, 7,k2 ] );
SetConjugate( FTL, 4, 3, [4,-1, 5,0, 6,0, 7,-k1 - 2*k2 + 2*k3 - 2*k4 ] );
SetConjugate( FTL, 5, 3, [4,0, 5,-1, 6,0, 7,-k1 - 2*k3 ] );
SetConjugate( FTL, 6, 3, [4,0, 5,0, 6,1, 7,0 ] );
SetConjugate( FTL, 7, 3, [4,0, 5,0, 6,0, 7,1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0, 7,2*k1 ] );
SetConjugate( FTL, 6, 4, [6,1, 7,0 ] );
SetConjugate( FTL, 6, 5, [6,1, 7,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr060 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 7 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [4,0, 5,0, 6,0, 7,k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 4,0, 5,-1, 6,1, 7,-2*k3 - k5 + 2*k6 ] );
SetConjugate( FTL, 3, 1, [3, 1 , 4,1, 5,0, 6,0, 7,k1 - k2 + k3 - k4 ] );
SetConjugate( FTL, 4, 1, [4,-1, 5,0, 6,0, 7,-2*(k1 - k2 + k3 - k4) ] );
SetConjugate( FTL, 5, 1, [4,0, 5,-1, 6,0, 7,-2*k3 ] );
SetConjugate( FTL, 6, 1, [4,0, 5,0, 6,-1, 7,2*(k3 + k5 - k6) ] );
SetConjugate( FTL, 7, 1, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [4,0, 5,1, 6,0, 7,k3 ] );
SetConjugate( FTL, 3, 2, [ 3, 1, 4,1, 5,1, 6,0, 7,2*k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 4, 2, [4,-1, 5,0, 6,0, 7,2*k1 ] );
SetConjugate( FTL, 5, 2, [4,0, 5,1, 6,0, 7,2*k3 ] );
SetConjugate( FTL, 6, 2, [4,0, 5,0, 6,-1, 7,0 ] );
SetConjugate( FTL, 7, 2, [4,0, 5,0, 6,0, 7,-1 ] );
SetRelativeOrder( FTL, 3, 2 );
SetPower( FTL, 3, [4,0, 5,0, 6,0, 7,k2 ] );
SetConjugate( FTL, 4, 3, [4,-1, 5,0, 6,0, 7,-2*(k1 - k2 + k3 - k4) ] );
SetConjugate( FTL, 5, 3, [4,0, 5,-1, 6,0, 7,2*(k1 - k3) ] );
SetConjugate( FTL, 6, 3, [4,0, 5,0, 6,1, 7,0 ] );
SetConjugate( FTL, 7, 3, [4,0, 5,0, 6,0, 7,1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0, 7,4*k1 ] );
SetConjugate( FTL, 6, 4, [6,1, 7,0 ] );
SetConjugate( FTL, 6, 5, [6,1, 7,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr061 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 7 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [4,0, 5,0, 6,0, 7,-k1 + k3 - k4 + k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 4,0, 5,-1, 6,1, 7,2*k1 + 2*k2 - k6 ] );
SetConjugate( FTL, 3, 1, [3, 1 , 4,1, 5,0, 6,-1, 7,-8*k1 - 2*k2 + 2*k3 - 2*k4 + k5 ] );
SetConjugate( FTL, 4, 1, [4,-1, 5,0, 6,0, 7,2*(4*k1 + k2 - k3 + k4) ] );
SetConjugate( FTL, 5, 1, [4,0, 5,-1, 6,0, 7,2*(k1 + k2 - k6) ] );
SetConjugate( FTL, 6, 1, [4,0, 5,0, 6,-1, 7,-2*(k1 + k2) ] );
SetConjugate( FTL, 7, 1, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [4,0, 5,1, 6,0, 7,k3 ] );
SetConjugate( FTL, 3, 2, [ 3, 1, 4,1, 5,1, 6,-1, 7,-6*k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 4, 2, [4,-1, 5,0, 6,0, 7,2*(3*k1 + k2 - k3 + k4) ] );
SetConjugate( FTL, 5, 2, [4,0, 5,1, 6,0, 7,0 ] );
SetConjugate( FTL, 6, 2, [4,0, 5,0, 6,-1, 7,-2*(k1 + k2) ] );
SetConjugate( FTL, 7, 2, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 3, 2 );
SetPower( FTL, 3, [4,0, 5,0, 6,1, 7,k2 ] );
SetConjugate( FTL, 4, 3, [4,-1, 5,0, 6,0, 7,2*k1 ] );
SetConjugate( FTL, 5, 3, [4,0, 5,-1, 6,0, 7,0 ] );
SetConjugate( FTL, 6, 3, [4,0, 5,0, 6,1, 7,2*k2 ] );
SetConjugate( FTL, 7, 3, [4,0, 5,0, 6,0, 7,-1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0, 7,0 ] );
SetConjugate( FTL, 6, 4, [6,1, 7,4*k1 ] );
SetConjugate( FTL, 6, 5, [6,1, 7,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr061b := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 7 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [4,0, 5,0, 6,0, 7,-k1 + k2 + k4 + k5 + k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 4,0, 5,-1, 6,1, 7,-2*k1 + 2*k2 - 2*k3 + 2*k4 + 2*k5 + k6 ] );
SetConjugate( FTL, 3, 1, [3, 1 , 4,1, 5,0, 6,-1, 7,2*k1 - 2*k2 + 2*k3 - 2*k4 - k5 ] );
SetConjugate( FTL, 4, 1, [4,-1, 5,0, 6,0, 7,-2*(k1 - k2 + k3 - k4) ] );
SetConjugate( FTL, 5, 1, [4,0, 5,-1, 6,0, 7,-2*k3 ] );
SetConjugate( FTL, 6, 1, [4,0, 5,0, 6,-1, 7,2*(k1 - k2 + k3 - k4 - k5) ] );
SetConjugate( FTL, 7, 1, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [4,0, 5,1, 6,0, 7,k3 ] );
SetConjugate( FTL, 3, 2, [ 3, 1, 4,1, 5,1, 6,-1, 7,2*k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 4, 2, [4,-1, 5,0, 6,0, 7,2*k1 ] );
SetConjugate( FTL, 5, 2, [4,0, 5,1, 6,0, 7,2*k3 ] );
SetConjugate( FTL, 6, 2, [4,0, 5,0, 6,-1, 7,0 ] );
SetConjugate( FTL, 7, 2, [4,0, 5,0, 6,0, 7,-1 ] );
SetRelativeOrder( FTL, 3, 2 );
SetPower( FTL, 3, [4,0, 5,0, 6,1, 7,k2 ] );
SetConjugate( FTL, 4, 3, [4,-1, 5,0, 6,0, 7,-2*(k1 - k2 + k3 - k4) ] );
SetConjugate( FTL, 5, 3, [4,0, 5,-1, 6,0, 7,2*(k1 - k3) ] );
SetConjugate( FTL, 6, 3, [4,0, 5,0, 6,1, 7,0 ] );
SetConjugate( FTL, 7, 3, [4,0, 5,0, 6,0, 7,1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0, 7,4*k1 ] );
SetConjugate( FTL, 6, 4, [6,1, 7,0 ] );
SetConjugate( FTL, 6, 5, [6,1, 7,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr061c:= function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 7 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [4,0, 5,0, 6,0, 7,k1 + k2 + k3 + k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 4,0, 5,-1, 6,1, 7,2*k2 + k6 ] );
SetConjugate( FTL, 3, 1, [3, 1 , 4,1, 5,0, 6,-1, 7,2*k1 + 2*k3 - k5 + 2*k6 ] );
SetConjugate( FTL, 4, 1, [4,-1, 5,0, 6,0, 7,-2*(k1 + k3 - k5 + k6) ] );
SetConjugate( FTL, 5, 1, [4,0, 5,-1, 6,0, 7,2*(k1 - k3) ] );
SetConjugate( FTL, 6, 1, [4,0, 5,0, 6,-1, 7,-2*k2 ] );
SetConjugate( FTL, 7, 1, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [4,0, 5,1, 6,0, 7,k3 ] );
SetConjugate( FTL, 3, 2, [ 3, 1, 4,1, 5,1, 6,-1, 7,-2*k2 + 2*k3 + k4 ] );
SetConjugate( FTL, 4, 2, [4,-1, 5,0, 6,0, 7,0 ] );
SetConjugate( FTL, 5, 2, [4,0, 5,1, 6,0, 7,2*k3 ] );
SetConjugate( FTL, 6, 2, [4,0, 5,0, 6,-1, 7,-2*k1 ] );
SetConjugate( FTL, 7, 2, [4,0, 5,0, 6,0, 7,-1 ] );
SetRelativeOrder( FTL, 3, 2 );
SetPower( FTL, 3, [4,0, 5,0, 6,1, 7,k2 ] );
SetConjugate( FTL, 4, 3, [4,-1, 5,0, 6,0, 7,0 ] );
SetConjugate( FTL, 5, 3, [4,0, 5,-1, 6,0, 7,2*k1 ] );
SetConjugate( FTL, 6, 3, [4,0, 5,0, 6,1, 7,2*k2 ] );
SetConjugate( FTL, 7, 3, [4,0, 5,0, 6,0, 7,-1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0, 7,0 ] );
SetConjugate( FTL, 6, 4, [6,1, 7,0 ] );
SetConjugate( FTL, 6, 5, [6,1, 7,4*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr062 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 7 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [4,0, 5,0, 6,0, 7,k3 - k4 + k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 4,0, 5,-1, 6,0, 7,-k6 ] );
SetConjugate( FTL, 3, 1, [3, 1 , 4,1, 5,0, 6,-1, 7,-3*k1 - 2*k2 + 2*k3 - 2*k4 + k5 ] );
SetConjugate( FTL, 4, 1, [4,-1, 5,0, 6,0, 7,3*k1 + 2*k2 - 2*k3 + 2*k4 ] );
SetConjugate( FTL, 5, 1, [4,0, 5,-1, 6,0, 7,-2*k6 ] );
SetConjugate( FTL, 6, 1, [4,0, 5,0, 6,-1, 7,-k1 - 2*k2 ] );
SetConjugate( FTL, 7, 1, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [4,0, 5,1, 6,0, 7,k3 ] );
SetConjugate( FTL, 3, 2, [ 3, 1, 4,1, 5,1, 6,-1, 7,-3*k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 4, 2, [4,-1, 5,0, 6,0, 7,3*k1 + 2*k2 - 2*k3 + 2*k4 ] );
SetConjugate( FTL, 5, 2, [4,0, 5,1, 6,0, 7,0 ] );
SetConjugate( FTL, 6, 2, [4,0, 5,0, 6,-1, 7,-k1 - 2*k2 ] );
SetConjugate( FTL, 7, 2, [4,0, 5,0, 6,0, 7,1 ] );
SetRelativeOrder( FTL, 3, 2 );
SetPower( FTL, 3, [4,0, 5,0, 6,1, 7,k2 ] );
SetConjugate( FTL, 4, 3, [4,-1, 5,0, 6,0, 7,k1 ] );
SetConjugate( FTL, 5, 3, [4,0, 5,-1, 6,0, 7,0 ] );
SetConjugate( FTL, 6, 3, [4,0, 5,0, 6,1, 7,2*k2 ] );
SetConjugate( FTL, 7, 3, [4,0, 5,0, 6,0, 7,-1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0, 7,0 ] );
SetConjugate( FTL, 6, 4, [6,1, 7,2*k1 ] );
SetConjugate( FTL, 6, 5, [6,1, 7,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr075 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 4 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,-1, 4,0, 5,k3 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr076 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 4 );
SetPower( FTL, 1, [2,0, 3,0, 4,1, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,-1, 4,0, 5,k3 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr077 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 4 );
SetPower( FTL, 1, [2,0, 3,0, 4,2, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,-1, 4,0, 5,k3 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr079 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 4 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,1, 4,1, 5,-k3 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,0, 4,-1, 5,k3 ] );
SetConjugate( FTL, 4, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,-k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr080 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 4 );
SetPower( FTL, 1, [2,1, 3,1, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,1, 4,1, 5,-k3 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,0, 4,-1, 5,k3 ] );
SetConjugate( FTL, 4, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,-k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr081 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 4 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k5 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,1, 4,0, 5,-k3 ] );
SetConjugate( FTL, 3, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k4 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr082 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 4 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k5 ] );
SetConjugate( FTL, 2, 1, [2,-1, 3,-1, 4,-1, 5,k2 + k3 + k4 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,0, 4,1, 5,-k4 ] );
SetConjugate( FTL, 4, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,0 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,-k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr083 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k2 + k3 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k2 + k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,k3 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr084 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,-1, 6,-k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k2 + k3 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k2 + k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,-2*k6 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,2, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,k3 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr085 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,-1, 4,0, 5,0, 6,k2 - k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,2*(k2 - k6) ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-2*k6 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k4 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,-k1 + k2 - 2*k6 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr086 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,-1, 4,0, 5,-1, 6,k2 - k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k1 + k2 + k3 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,k1 - k2 + k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,-k1 + k2 - k3 - 2*k6 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,2, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,k3 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr087 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-2*k2 + 2*k3 + k5 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,2*k2 - k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,1, 5,1, 6,-k3 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 5, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,-k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr088 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,0, 5,0, 6,-k2 - k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,2*(k2 + k6) ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k1 + 2*k3 + 2*k6 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,-2*k6 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,-1, 4,-1, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,1, 5,1, 6,-k3 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,0, 5,-1, 6,k3 ] );
SetConjugate( FTL, 5, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,-k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr103 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 3 , 3,0, 4,0, 5,0, 6,-k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,-k2 - k3 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,k3 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr104 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,1, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 3 , 3,1, 4,0, 5,0, 6,-k2 - 2*k3 - k5 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,-2*(k2 + k3 + k5) ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k1 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*(k2 + k3 + k4 + k5) ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,-k1 + k2 + 2*k3 + 2*k5 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr106 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 3 , 3,1, 4,0, 5,-1, 6,-k3 - k4 + k5 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,2*k5 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k1 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,-2*k4 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,2, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,-1, 5,0, 6,-k1 - k2 - 2*k5 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr110 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,1, 5,0, 6,k1 - k2 - k4 ] );
SetConjugate( FTL, 2, 1, [2, 3 , 3,1, 4,1, 5,1, 6,-2*k1 + 2*k2 - 2*k3 + 2*k4 - k5 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,0, 5,-1, 6,-k4 ] );
SetConjugate( FTL, 4, 1, [3,1, 4,1, 5,1, 6,2*k1 - 2*k2 - k4 ] );
SetConjugate( FTL, 5, 1, [3,-1, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,-1, 4,-1, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,1, 5,1, 6,-k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,0, 5,-1, 6,k2 ] );
SetConjugate( FTL, 5, 2, [3,-1, 4,0, 5,0, 6,4*k1 - 4*k2 + 2*k3 - 3*k4 + 2*k5 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,0 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,-2*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr114 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,1, 5,0, 6,k4 ] );
SetConjugate( FTL, 2, 1, [2, 3 , 3,0, 4,1, 5,1, 6,-k1 + k2 - 2*k3 + 2*k4 - k5 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k1 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,1, 5,0, 6,2*k4 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,0 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 4 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,1, 5,0, 6,-k1 + k2 + 2*k4 ] );
SetConjugate( FTL, 4, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,-1, 6,2*(k1 - k2 + k3 - k4 + k5) ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr143 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 3 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,-1, 3,-1, 4,0, 5,k2 + k3 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr144 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 3 );
SetPower( FTL, 1, [2,0, 3,0, 4,1, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,-1, 3,-1, 4,0, 5,k2 + k3 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr146 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 3 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,0, 4,1, 5,k2 + k3 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,1, 4,0, 5,-k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,-k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr147 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 6 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k5 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,1, 4,0, 5,k1 - k2 - k3 ] );
SetConjugate( FTL, 3, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k4 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr148 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 6 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k5 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,0, 4,-1, 5,k4 ] );
SetConjugate( FTL, 3, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,-1, 4,0, 5,k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,-k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr158 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 2 , 3,0, 4,0, 5,0, 6,-k4 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,-1, 5,0, 6,-k2 ] );
SetConjugate( FTL, 4, 1, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 3 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,-1, 5,0, 6,k2 + k3 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr159 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 2 , 3,0, 4,0, 5,0, 6,-k3 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,1, 5,0, 6,-k4 ] );
SetConjugate( FTL, 4, 1, [3,1, 4,0, 5,0, 6,-k4 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 3 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,-1, 5,0, 6,k1 - k2 + 3*k4 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k1 + 2*k2 - 3*k4 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr161 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 2 , 3,-1, 4,0, 5,0, 6,-k3 - k4 + 2*k5 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,1, 5,0, 6,-k1 - k2 - 2*k3 + 2*k5 ] );
SetConjugate( FTL, 4, 1, [3,1, 4,0, 5,0, 6,-k2 - 2*k3 + 2*k5 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 3 );
SetPower( FTL, 2, [3,1, 4,1, 5,1, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,0, 4,0, 5,1, 6,k2 + k3 ] );
SetConjugate( FTL, 4, 2, [3,1, 4,0, 5,0, 6,-k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,1, 5,0, 6,-k3 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,-k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr168 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 6 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,1, 4,0, 5,k1 - k2 - k3 ] );
SetConjugate( FTL, 3, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr169 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 6 );
SetPower( FTL, 1, [2,0, 3,0, 4,5, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,1, 4,0, 5,k1 - k2 - k3 ] );
SetConjugate( FTL, 3, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr172 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 6 );
SetPower( FTL, 1, [2,0, 3,0, 4,2, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,1, 4,0, 5,k1 - k2 - k3 ] );
SetConjugate( FTL, 3, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr173 := function (k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 6 );
SetPower( FTL, 1, [2,0, 3,0, 4,3, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,1, 4,0, 5,k1 - k2 - k3 ] );
SetConjugate( FTL, 3, 1, [2,-1, 3,0, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,0 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr174 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 6 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k5 ] );
SetConjugate( FTL, 2, 1, [2,-1, 3,-1, 4,0, 5,k2 + k3 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k4 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,0, 5,k1 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr175 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k6 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k1 - 2*k3 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k1 + 2*k2 + 2*k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 6 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,1, 5,0, 6,k1 - k2 - k3 ] );
SetConjugate( FTL, 4, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr176 := function (k1, k2, k3 , k4, k5, k6)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,0, 5,-1, 6,k6 ] );
SetConjugate( FTL, 3, 1, [3,-1, 4,0, 5,0, 6,k1 - 2*k3 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,-k1 + 2*k2 + 2*k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,2*k6 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 6 );
SetPower( FTL, 2, [3,0, 4,0, 5,3, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,1, 5,0, 6,k1 - k2 - k3 ] );
SetConjugate( FTL, 4, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4Nr184 := function (k1, k2, k3 , k4, k5)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,1, 6,k5 ] );
SetConjugate( FTL, 2, 1, [2, 5 , 3,0, 4,0, 5,0, 6,-k4 ] );
SetConjugate( FTL, 3, 1, [3,0, 4,-1, 5,0, 6,-k1 + k2 + 2*k3 ] );
SetConjugate( FTL, 4, 1, [3,-1, 4,0, 5,0, 6,k1 - k2 - 2*k3 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,1, 6,2*k5 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 6 );
SetPower( FTL, 2, [3,0, 4,0, 5,0, 6,k4 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,1, 5,0, 6,k1 - k2 - k3 ] );
SetConjugate( FTL, 4, 2, [3,-1, 4,0, 5,0, 6,k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,0 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB1 := function (k , k1, k2 , k3)
local FTL;
FTL := FromTheLeftCollector( 4 );
SetConjugate( FTL, 2, 1, [2,1, 3, k, 4, k1 ] );
SetConjugate( FTL, 3, 1, [3,1, 4, k2 ] );
SetConjugate( FTL, 3, 2, [3,1, 4, k3 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB2 := function (k, k1, k2, k3)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,1, 5,k3 ] );
SetConjugate( FTL, 2, 1, [2,-1, 3,0, 4,0, 5,k1 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,1, 5,2*k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,-1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,2*k, 5,2*(k*k1 + k*k2 + k*k3) ] );
SetConjugate( FTL, 4, 2, [4,1, 5,2*k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,2*k2 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB3c := function (l, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,0, 5,0 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,0, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,2*l, 5,(k1 - k3)*l ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB3b := function (l, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,-1, 5,-k2 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,0, 5,k3 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,-2*k2 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,2*l, 5,-k1 + 2*k1*l + 2*k2*l ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,2*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB3 := function (l, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,-1, 5,-k2 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,0, 5,k3 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,-2*k2 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,1 + 2*l, 5,k2 + k1*l + 2*k2*l ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB4 := function (k, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,1, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,0, 5,0 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,-k, 5,k*k1 + k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,2*k, 5,k*(k1 - k3) ] );
SetConjugate( FTL, 4, 2, [4,1, 5,0 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB4b := function (k, k1, k2, k3)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,1, 3,0, 4,0, 5,k3 ] );
SetConjugate( FTL, 2, 1, [2,1, 3,0, 4,0, 5,2*k3 ] );
SetConjugate( FTL, 3, 1, [2,0, 3,-1, 4,-k, 5,k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k1 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,-1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,2*k, 5,-(k*k1) - 2*k2 ] );
SetConjugate( FTL, 4, 2, [4,1, 5,-2*k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,0 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB5 := function (l, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,1, 4,0, 5,k2 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,2*l, 5,-(k3*l) ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,-k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB5b := function (l, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 5 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [2,0, 3,0, 4,0, 5,k4 ] );
SetConjugate( FTL, 2, 1, [2,0, 3,1, 4,0, 5,k2 ] );
SetConjugate( FTL, 3, 1, [2,1, 3,0, 4,0, 5,-k2 ] );
SetConjugate( FTL, 4, 1, [2,0, 3,0, 4,-1, 5,2*k3 ] );
SetConjugate( FTL, 5, 1, [2,0, 3,0, 4,0, 5,1 ] );
SetConjugate( FTL, 3, 2, [3,1, 4,1 + 2*l, 5,-(k3*(1 + 2*l)) ] );
SetConjugate( FTL, 4, 2, [4,1, 5,k1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,-k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB7 := function (l, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,-1, 5,-1, 6,2*k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,2*l, 6,-((k1 - 2*k2)*l) ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,k1 - 2*k2 + 2*k3 - 2*k4 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k1 - 2*k2 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,k1 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,4*l, 6,4*(k1*l + k2*l) ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,2*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB7b := function (l, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,0, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,0, 4,-1, 5,-1, 6,4*k1 - 2*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,1 + 2*l, 6,-k1 + k2 - 2*k1*l + 2*k2*l ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,0, 6,2*(k1 - k2 + k3 - k4) ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,2*(k1 - k2) ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,0, 6,2*k1 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,2*(1 + 2*l), 6,2*(2*k1 + k2 + 4*k1*l + 2*k2*l) ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,4*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB8 := function (k, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,-1, 5,-1 - 4*k, 6,2*k1 + 8*k*k1 - 2*k2 - 6*k*k2 + 2*k3 - k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,0 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,-2*k, 6,k1 + 6*k*k1 - 2*k2 - 6*k*k2 + 2*k3 - 2*k4 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k1 - 2*k2 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,2*k, 6,2*k*k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,-2*k, 6,k1 + 2*k*k1 - 2*k*k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,4*k, 6,2*(k*k1 + 2*k*k2) ] );
SetConjugate( FTL, 5, 3, [5,1, 6,0 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,2*k1 ] );
return PcpGroupByCollector(FTL);
end;
ACPcpGroupDim4NrB8b := function (k, k1, k2, k3 , k4)
local FTL;
FTL := FromTheLeftCollector( 6 );
SetRelativeOrder( FTL, 1, 2 );
SetPower( FTL, 1, [3,1, 4,0, 5,0, 6,k3 ] );
SetConjugate( FTL, 2, 1, [2, 1 , 3,1, 4,-1, 5,-1 - 4*k, 6,k1 + 2*k*k1 - 2*k*k2 + 2*k3 + k4 ] );
SetConjugate( FTL, 3, 1, [3,1, 4,0, 5,0, 6,2*k3 ] );
SetConjugate( FTL, 4, 1, [3,0, 4,-1, 5,-2*k, 6,-2*k*k2 ] );
SetConjugate( FTL, 5, 1, [3,0, 4,0, 5,-1, 6,k1 ] );
SetConjugate( FTL, 6, 1, [3,0, 4,0, 5,0, 6,-1 ] );
SetRelativeOrder( FTL, 2, 2 );
SetPower( FTL, 2, [3,0, 4,0, 5,1, 6,k2 ] );
SetConjugate( FTL, 3, 2, [3,-1, 4,0, 5,2*k, 6,-k1 + 2*k*k1 + 2*k*k2 ] );
SetConjugate( FTL, 4, 2, [3,0, 4,-1, 5,-2*k, 6,-2*k*k2 ] );
SetConjugate( FTL, 5, 2, [3,0, 4,0, 5,1, 6,2*k2 ] );
SetConjugate( FTL, 6, 2, [3,0, 4,0, 5,0, 6,-1 ] );
SetConjugate( FTL, 4, 3, [4,1, 5,4*k, 6,-2*(k*k1 - 2*k*k2) ] );
SetConjugate( FTL, 5, 3, [5,1, 6,-2*k1 ] );
SetConjugate( FTL, 5, 4, [5,1, 6,0 ] );
return PcpGroupByCollector(FTL);
end;
#############################################################################
##
## some small helpers
##
ACPcpDim4Funcs := [ ACPcpGroupDim4Nr001, ACPcpGroupDim4Nr002,
ACPcpGroupDim4Nr003,
ACPcpGroupDim4Nr004, ACPcpGroupDim4Nr004b, ACPcpGroupDim4Nr005,
ACPcpGroupDim4Nr006, ACPcpGroupDim4Nr007, ACPcpGroupDim4Nr007b,
ACPcpGroupDim4Nr008, ACPcpGroupDim4Nr009, ACPcpGroupDim4Nr009b,
ACPcpGroupDim4Nr010, ACPcpGroupDim4Nr011, ACPcpGroupDim4Nr012,
ACPcpGroupDim4Nr013, ACPcpGroupDim4Nr014, ACPcpGroupDim4Nr014b,
ACPcpGroupDim4Nr015, ACPcpGroupDim4Nr018, ACPcpGroupDim4Nr019,
ACPcpGroupDim4Nr019b, ACPcpGroupDim4Nr019c, ACPcpGroupDim4Nr026,
ACPcpGroupDim4Nr027, ACPcpGroupDim4Nr029, ACPcpGroupDim4Nr029b,
ACPcpGroupDim4Nr029c, ACPcpGroupDim4Nr030, ACPcpGroupDim4Nr031,
ACPcpGroupDim4Nr032, ACPcpGroupDim4Nr033, ACPcpGroupDim4Nr033b,
ACPcpGroupDim4Nr033c, ACPcpGroupDim4Nr034, ACPcpGroupDim4Nr036,
ACPcpGroupDim4Nr037, ACPcpGroupDim4Nr041, ACPcpGroupDim4Nr043,
ACPcpGroupDim4Nr045, ACPcpGroupDim4Nr055, ACPcpGroupDim4Nr056,
ACPcpGroupDim4Nr058, ACPcpGroupDim4Nr060, ACPcpGroupDim4Nr061,
ACPcpGroupDim4Nr061b, ACPcpGroupDim4Nr061c, ACPcpGroupDim4Nr062,
ACPcpGroupDim4Nr075, ACPcpGroupDim4Nr076, ACPcpGroupDim4Nr077,
ACPcpGroupDim4Nr079, ACPcpGroupDim4Nr080, ACPcpGroupDim4Nr081,
ACPcpGroupDim4Nr082, ACPcpGroupDim4Nr083, ACPcpGroupDim4Nr084,
ACPcpGroupDim4Nr085, ACPcpGroupDim4Nr086, ACPcpGroupDim4Nr087,
ACPcpGroupDim4Nr088, ACPcpGroupDim4Nr103, ACPcpGroupDim4Nr104,
ACPcpGroupDim4Nr106, ACPcpGroupDim4Nr110, ACPcpGroupDim4Nr114,
ACPcpGroupDim4Nr143, ACPcpGroupDim4Nr144, ACPcpGroupDim4Nr146,
ACPcpGroupDim4Nr147, ACPcpGroupDim4Nr148, ACPcpGroupDim4Nr158,
ACPcpGroupDim4Nr159, ACPcpGroupDim4Nr161, ACPcpGroupDim4Nr168,
ACPcpGroupDim4Nr169, ACPcpGroupDim4Nr172, ACPcpGroupDim4Nr173,
ACPcpGroupDim4Nr174, ACPcpGroupDim4Nr175, ACPcpGroupDim4Nr176,
ACPcpGroupDim4Nr184, ACPcpGroupDim4NrB1, ACPcpGroupDim4NrB2,
ACPcpGroupDim4NrB3c, ACPcpGroupDim4NrB3b, ACPcpGroupDim4NrB3,
ACPcpGroupDim4NrB4, ACPcpGroupDim4NrB4b, ACPcpGroupDim4NrB5,
ACPcpGroupDim4NrB5b, ACPcpGroupDim4NrB7, ACPcpGroupDim4NrB7b,
ACPcpGroupDim4NrB8, ACPcpGroupDim4NrB8b ];
MakeReadOnlyGlobal( "ACPcpDim4Funcs" );
aclib-1.3.2/doc/intro.tex 0000664 0003717 0003717 00000012560 13627151544 016437 0 ustar gap-jenkins gap-jenkins %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%W aclib.tex Karel Dekimpe
%W Bettina Eick
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Chapter{The Almost Crystallographic Groups Package}
A group is called *almost crystallographic* if it is a finitely generated
nilpotent-by-finite group without non-trivial finite normal subgroups. An
important special case of almost crystallographic groups are the *almost
Bieberbach groups*: these are almost crystallographic and torsion free.
By its definition, an almost crystallographic group $G$ has a finitely
generated nilpotent normal subgroup $N$ of finite index. Clearly, $N$ is
polycyclic and thus has a polycyclic series. The number of infinite cyclic
factors in such a series for $N$ is an invariant of $G$: the *Hirsch length*
of $G$.
For each almost crystallographic group of Hirsch length 3 and 4 there exists
a representation as a rational matrix group in dimension 4 or 5, respectively.
These representations can be considered as affine representations of dimension
3 or 4. Via these representations, the almost crystallographic groups act
(properly discontinuously) on $\R^3$ or $\R^4$. That is one reason to define
the *dimension* of an almost crystallographic group as its Hirsch length.
The 3-dimensional and a part of the 4-dimensional almost crystallographic
groups have been classified by K. Dekimpe in \cite{KD}. This classification
includes all almost Bieberbach groups in dimension 3 and 4. It is the first
central aim of this package to give access to the resulting library of groups.
The groups in this electronic catalog are available in two different
representations: as rational matrix groups and as polycyclically presented
groups. While the first representation is the more natural one, the latter
description facilitates effective computations with the considered groups
using the methods of the {\sf Polycyclic} package.
The second aim of this package is to introduce a variety of algorithms for
computations with polycyclically presented almost crystallographic groups.
These algorithms supplement the methods available in the {\sf Polycyclic}
package and give access to some methods which are interesting specifically
for almost crystallographic groups. In particular, we present methods to
compute Betti numbers and to construct or check the existence of certain
extensions of almost crystallographic groups. We note that these methods
have been applied in \cite{DE1} and \cite{DE2} for computations with
almost crystallographic groups.
Finally, we remark that almost crystallographic groups can be seen as natural
generalizations of crystallographic groups. A library of crystallographic
groups and algorithms to compute with crystallographic groups are available
in the \GAP\ packages `cryst', `carat' and `crystcat'.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{More about almost crystallographic groups}
Almost crystallographic groups were first discussed in the theory of
actions on Lie groups. We recall the original definition here briefly
and we refer to \cite{AUS}, \cite{KD} and \cite{LEE} for more details.
Let $L$ be a connected and simply connected nilpotent Lie group. For
example, the 3-dimensional Heisenberg group, consisting of all upper
unitriangular $3\times3$--matrices with real entries is of this type.
Then $L\rtimes {\rm Aut}(L)$ acts affinely (on the left) on $L$ via
$$ \forall l,l'\in L,\forall \alpha \in {\rm Aut}(L):\;
^{(l,\alpha)}l'=l \, \alpha(l'). $$
Let $C$ be a maximal compact subgroup of ${\rm Aut}(L)$. Then a subgroup $G$
of $L \rtimes C$ is said to be an almost crystallographic group if and only
if the action of $G$ on $L$, induced by the action of $L\rtimes {\rm Aut}(L)$,
is properly discontinuous and the quotient space $G \backslash L$ is compact.
One recovers the situation of the ordinary crystallographic groups by taking
$L={\Bbb R}^n$, for some $n$, and $C=O(n)$, the orthogonal group.
More generally, we say that an abstract group is an almost crystallographic
group if it can be realized as a genuine almost crystallographic subgroup
of some $L \rtimes C$. In the following theorem we outline some algebraic
characterizations of almost crystallographic groups; see Theorem 3.1.3 of
\cite{KD}. Recall that the *Fitting subgroup Fitt$(G)$* of a
polycyclic-by-finite group $G$ is its unique maximal normal nilpotent
subgroup.
*Theorem.*
The following are equivalent for a polycyclic-by-finite group $G$:
\beginlist
\item{(1)} $G$ is an almost crystallographic group.
\item{(2)} Fitt$(G)$ is torsion free and of finite index in $G$.
\item{(3)} $G$ contains a torsion free nilpotent normal subgroup $N$
of finite index in $G$ with $C_G(N)$ torsion free.
\item{(4)} $G$ has a nilpotent subgroup of finite index and there
are no non-trivial finite normal subgroups in $G$.
\endlist
In particular, if $G$ is almost crystallographic, then $G / Fitt(G)$
is finite. This factor is called the *holonomy group* of $G$.
The dimension of an almost crystallographic group equals the dimension
of the Lie group $L$ above which coincides also with the Hirsch length
of the polycyclic-by-finite group. This library therefore contains
families of virtually nilpotent groups of Hirsch length 3 and 4.
aclib-1.3.2/doc/aclib.bib 0000664 0003717 0003717 00000003576 13627151544 016321 0 ustar gap-jenkins gap-jenkins %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%W aclib.bib ac lib documentation Bettina Eick
%W Karel Dekimpe
%%
@book{KD,
author = "Karel Dekimpe",
title = "Almost-Bieberbach Groups: Affine and Polynomial Structures",
series = "Lecture Notes in Mathematics",
volume = "1639",
publisher = "Springer-Verlag, Heidelberg",
year = "1996",
}
@article{AUS,
author = "Louis Auslander",
title = "Bieberbach's theorem on space groups and discrete uniform
subgroups of Lie groups",
journal = "Ann. of Math",
number = "3",
volume = "71",
year = "1960",
pages ="579-590"
}
@book{BRO,
author = "Kenneth S.\ Brown",
title = "Cohomology of Groups",
volume = "87",
series = "Grad. Texts in Math.",
year = "1982",
publisher = "Springer-Verlag, New York"
}
@article{LEE,
author = "Kyung Bai Lee",
title = "There are only finitely many infra-nilmanifolds under
each nilmanifold",
journal = "Quart. J. Math. Oxford",
number = "2",
volume = "39",
year = "1988",
pages ="61--66"
}
@article{DM,
author = "An Descheemaeker and Wim Malfait",
title = "{$P$}-localization of Relative Groups",
journal = "Journal of Pure and Applied Algebra",
volume = "159",
number = "1",
year = "2001"
}
@article{DE1,
author = "Karel Dekimpe and Bettina Eick",
title = "Computations with almost crystallographic groups",
journal = "Submitted",
year = "2001" }
@article{DE2,
author = "Karel Dekimpe and Bettina Eick",
title = "Computational aspects of group extensions and their
application in topology",
journal = "Submitted",
year = "2001" }
aclib-1.3.2/doc/algos.tex 0000664 0003717 0003717 00000010425 13627151544 016407 0 ustar gap-jenkins gap-jenkins \Chapter{Algorithms for almost crystallographic groups}
This chapter presents a variety of algorithms for almost crystallographic
groups. In most cases, they assume a polycyclically presented group as
input; in particular, the input groups must be polycyclic in this case.
The methods described here supplement the methods of the {\sf Polycyclic}
package for polycyclically presented groups. Many of the functions in this
chapter are based on methods of the {\sf Polycyclic} package and thus this
package must be installed to use the functions introduced here. We refer to
the {\sf Polycyclic} package for further information on polycyclic
presentations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Properties of almost crystallographic groups}
\> IsAlmostCrystallographic( ) P
This function checks if a polycyclically presented group is almost
crystallographic; that is, it checks if is nilpotent-by-finite and
has no non-trivial finite normal subgroup.
\> IsAlmostBieberbachGroup( ) P
This function checks if a polycyclically presented group is almost
Bieberbach; that is, it checks if is nilpotent-by-finite and torsion
free.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Betti numbers}
Let $G$ be a polycyclically presented and torsion free group of Hirsch
length $n$. Then we can compute the Betti numbers $\beta_i(G)$ for $i \in
\{0, 1, 2, n-2, n-1, n\}$. If $n \leq 6$, then we can compute all Betti
numbers $\beta_i(G)$ for $0 \leq i \leq 6$ of $G$. We introduce the following
functions for this purpose and we refer to \cite{BRO} for the details on
the orientation module and the Betti numbers.
\> OrientationModule( ) F
This function determines the orientation module of the polycyclically
presented group ; that is, it returns a list of matrices $m_1, \ldots,
m_n \leq GL( 1, \Z )$ which are the images of the 'Igs(G)' in their action
on the orientation module.
\> BettiNumber( , ) F
This function returns the th Betti number of the polycyclically presented
torsion free group if $m \in \{0, 1, 2, n-2, n-1, n\}$, where $n$ is the
Hirsch length of .
\> BettiNumbers( ) A
This function returns the Betti number of the polycyclically presented
torsion free group if the Hirsch length of is smaller than 7.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Determination of certain extensions}
Let $G$ be a polycyclically presented almost crystallographic group. We want
to check the existence of certain extensions of $G$.
First, it is well-known that the equivalence classes of extensions of $G$
correspond to the second cohomology group of $G$. This cohomology group can
be computed using the methods of the {\sf Polycyclic} package for any
explicitly given module of $G$. Further, we can construct a polycyclic
presentation for each cocycle of the second cohomology group. We give an
example for such a computation below.
However, we may be interested in certain extensions only; for example,
the torsion free extensions are often of particular interest. If the
second cohomology group is finite, then we can compute a polycyclic
presentation for each element of this group and check the resulting group
for torsion freeness. But if the second cohomology group is infinite, then
this approach is not available. Hence we introduce the following special
method to cover this and related applications.
\> HasExtensionOfType( , , ) F
Suppose that is a polycyclically presented almost crystallographic group
with Fitting subgroup $N$. This function checks if there is a $G$-module
$M \cong \Z$ which is centralized by $N$ such that there exists a torsion
free extension of $M$ by (if the flag is true) or an
extension $E$ with $Z(Fitt(E)) = M$ (if the flag is true)
or an extension which satisfies both conditions (if both flags are true).
We note that the existence of such extensions is of interest in the
determination of extensions which are almost Bieberbach groups. We refer
to \cite{DE1} for a more detailed account of this application and for
further results of a similar nature.
aclib-1.3.2/doc/examp.tex 0000664 0003717 0003717 00000021715 13627151544 016420 0 ustar gap-jenkins gap-jenkins \Chapter{Example computations with almost crystallographic groups}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Example computations I}
Using the functions available for pcp groups in the share package
{\sf polycyclic} it is now easy to redo some of the calculations of
\cite{KD}. As a first example we check whether the groups indicated
as torsion free in \cite{KD} are also determined as torsion free
ones by \GAP. In \cite{KD} these almost Bieberbach groups are listed as
``AB-groups''. So for type ``013'' these are the groups with parameters
$(k,0,1,0,1,0)$ where $k$ is an even integer. Let's look at some examples
in \GAP:
\beginexample
gap> G:=AlmostCrystallographicPcpDim4("013",[8,0,1,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
true
gap> G:=AlmostCrystallographicPcpDim4("013",[9,0,1,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
false
\endexample
Further, there is also some cohomology information in the tables
of \cite{KD}. In fact, the groups in this library were obtained
as extensions $E$ of the form
$$
1 \rightarrow \Z \rightarrow E \rightarrow Q \rightarrow 1
$$
where, in the 4-dimensional case $Q = E/\langle d \rangle$. The
cohomology information for the particular example above shows that
the groups determined by a parameter set $(k_1,k_2,k_3,k_4,k_4,k_6)$
are equivalent as extensions to the groups determined by the parameters
$(k_1, k_2 \bmod 2, k_3 \bmod 2, k_4 \bmod 2, k_5 \bmod 2, 0)$. This is
also visible in finding torsion:
\beginexample
gap> G:=AlmostCrystallographicPcpDim4("013",[10,0,2,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
false
gap> G:=AlmostCrystallographicPcpDim4("013",[10,0,3,0,1,9]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
true
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Example computations II}
The computation of cohomology groups played an important role in the
classification of the almost Bieberbach groups in \cite{KD}. Using
\GAP, it is now possible to check these computations. As an example we
consider the 4-dimensional almost crystallographic groups of type 85 on
page 202 of \cite{KD}. This group $E$ has 6 generators. In the table, one
also finds the information
$$
H^2(Q,\Z) = \Z \oplus (\Z_2)^2 \oplus \Z_4
$$
for $Q=E/\langle d \rangle$ as above. Moreover, the $Q$--module $\Z$ is
in fact the group $\langle d \rangle$, where the $Q$-action comes from
conjugation inside $E$. In the case of groups of type 85, $\Z$ is a
trivial $Q$-module. The following example demonstrates how to (re)compute
this two-cohomology group $H^2(Q,\Z)$.
\beginexample
gap> G:=AlmostCrystallographicPcpGroup(4, "085", false);
Pcp group with orders [ 2, 4, 0, 0, 0, 0 ]
gap> GroupGeneratedByd:=Subgroup(G, [G.6] );
Pcp group with orders [ 0 ]
gap> Q:=G/GroupGeneratedByd;
Pcp group with orders [ 2, 4, 0, 0, 0 ]
gap> action:=List( Pcp(Q), x -> [[1]] );
[ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ]
gap> C:=CRRecordByMats( Q, action);;
gap> TwoCohomologyCR( C ).factor.rels;
[ 2, 2, 4, 0 ]
\endexample
This last line gives us the abelian invariants of the second
cohomology group $H^2(Q,\Z)$. So we should read this line as
$$
H^2(Q,\Z) = \Z_2 \oplus \Z_2 \oplus \Z_4 \oplus \Z
$$
which indeed coincides with the information in \cite{KD}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Example computations III}
As another application of the capabilities of the combination of
`aclib' and {\sf polycyclic} we check some computations of \cite{DM}.
Section 5 of the paper \cite{DM} is completely devoted to an example
of the computation of the $P$-localization of a virtually nilpotent group,
where $P$ is a set of primes. Although it is not our intention to
develop the theory of $P$-localization of groups at this place, let us
summarize some of the main results concerning this topic here.
For a set of primes $P$, we say that $n \in P$ if and only if $n$ is
a product of primes in $P$. A group $G$ is said to be $P$-local if and
only if the map $\mu_n:G\rightarrow G: g \mapsto g^n$ is bijective for
all $n \in P'$, where $P'$ is the set of all primes not in $P$. The
$P$-localization of a group $G$, is a $P$-local group $G_P$ together
with a morphism $\alpha :G \rightarrow G_P$ which satisfy the following
universal property: For each $P$-local group $L$ and any morphism
$\varphi: G \rightarrow L$, there exists a unique morphism $\psi:G_P
\rightarrow L$, such that $\psi \circ \alpha = \varphi$.
This concept of localization is well developed for finite groups and
for nilpotent groups. For a finite group $G$, the $P$-localization is
the largest quotient of $G$, having no elements with an order belonging to
$P'$ (the morphism $\alpha$, mentioned above is the natural projection).
In \cite{DM} a contribution is made towards the localization of virtually
nilpotent groups. The theory developed in the paper is then illustrated
in the last section of the paper by means of the computation of the
$P$-localization of an almost crystallographic group. For their example
the authors have chosen an almost crystallographic group $G$ of dimension 3
and type 17. For the set of parameters $(k_1,k_2,k_3,k_4)$ they have
considered all cases of the form $(k_1,k_2,k_3,k_4)=(2,0,0,k_4)$.
Here we will check their computations in two cases $k_4=0$ and $k_4=1$
using the set of primes $P=\{2\}$. The holonomy group of these almost
crystallographic groups $G$ is the dihedral group ${\cal D}_6$ of order
12. Thus there is a short exact sequence of the form
$$ 1 \rightarrow {\rm Fitt}(G) \rightarrow G
\rightarrow {\cal D}_6 \rightarrow 1. $$
As a first step in their computation, Descheemaeker and Malfait determine
the group $I_{P'}{\cal D}_6$, which is the unique subgroup of order 3 in
${\cal D}_6$. One of the main objects in \cite{DM} is the group $K=p^{-1}
(I_{P'}{\cal D}_6)$, where $p$ is the natural projection of $G$ onto its
holonomy group. It is known that the $P$-localization of $G$ coincides
with the $P$-localization of $G/\gamma_3(K)$, where $\gamma_3(K)$ is the
third term in the lower central series of $K$. As $G/\gamma_3(K)$ is
finite in this example, we exactly know what this $P$-localization is.
Let us now show, how GAP can be used to compute this $P$-localization in
two cases:
\medskip
First case: The parameters are $(k_1,k_2,k_3,k_4)=(2,0,0,0)$
\beginexample
gap> G := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,0] );
Pcp group with orders [ 2, 6, 0, 0, 0 ]
gap> projection := NaturalHomomorphismOnHolonomyGroup( G );
[ g1, g2, g3, g4, g5 ] -> [ g1, g2, identity, identity, identity ]
gap> F := HolonomyGroup( G );
Pcp group with orders [ 2, 6 ]
gap> IPprimeD6 := Subgroup( F , [F.2^2] );
Pcp group with orders [ 3 ]
gap> K := PreImage( projection, IPprimeD6 );
Pcp group with orders [ 3, 0, 0, 0 ]
gap> PrintPcpPresentation( K );
pcp presentation on generators [ g2^2, g3, g4, g5 ]
g2^2 ^ 3 = identity
g3 ^ g2^2 = g3^-1*g4^-1
g3 ^ g2^2^-1 = g4*g5^-2
g4 ^ g2^2 = g3*g5^2
g4 ^ g2^2^-1 = g3^-1*g4^-1*g5^2
g4 ^ g3 = g4*g5^2
g4 ^ g3^-1 = g4*g5^-2
gap> Gamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));
Pcp group with orders [ 0, 0, 0 ]
gap> quotient := G/Gamma3K;
Pcp group with orders [ 2, 6, 3, 3, 2 ]
gap> S := SylowSubgroup( quotient, 3);
Pcp group with orders [ 3, 3, 3 ]
gap> N := NormalClosure( quotient, S);
Pcp group with orders [ 3, 3, 3 ]
gap> localization := quotient/N;
Pcp group with orders [ 2, 2, 2 ]
gap> PrintPcpPresentation( localization );
pcp presentation on generators [ g1, g2, g3 ]
g1 ^ 2 = identity
g2 ^ 2 = identity
g3 ^ 2 = identity
\endexample
This shows that $G_P\cong \Z_2^3$.
\medskip
Second case: The parameters are $(k_1,k_2,k_3,k_4)=(2,0,0,1)$
\beginexample
gap> G := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,1]);;
gap> projection := NaturalHomomorphismOnHolonomyGroup( G );;
gap> F := HolonomyGroup( G );;
gap> IPprimeD6 := Subgroup( F , [F.2^2] );;
gap> K := PreImage( projection, IPprimeD6 );;
gap> Gamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));;
gap> quotient := G/Gamma3K;;
gap> S := SylowSubgroup( quotient, 3);;
gap> N := NormalClosure( quotient, S);;
gap> localization := quotient/N;
Pcp group with orders [ 2, 2, 2 ]
gap> PrintPcpPresentation( localization );
pcp presentation on generators [ g1, g2, g3 ]
g1 ^ 2 = identity
g2 ^ 2 = g3
g3 ^ 2 = identity
g2 ^ g1 = g2*g3
g2 ^ g1^-1 = g2*g3
\endexample
In this case, we see that $G_P={\cal D}_4$.
\medskip
The reader can check that these results coincide with those obtained in
\cite{DM}. Note also that we used a somewhat different scheme to compute
this localization than the one used in \cite{DM}. We invite the reader to
check the same computations, tracing exactly the steps made in \cite{DM}.
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\centerline{\titlefont ---}\vfill
\centerline{\titlefont A GAP4 Package} \vfill
\centerline{\secfont Computations with} \bigskip
\centerline{\secfont Almost Crystallographic Groups} \vfill
\centerline{\secfont by}\vfill
\centerline{\secfont Karel Dekimpe (KU Leuven Kulak)}\medskip
\centerline{\secfont and }\medskip
\centerline{\secfont Bettina Eick (Braunschweig)}
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aclib-1.3.2/doc/aclib.tex 0000664 0003717 0003717 00000031142 13627151544 016353 0 ustar gap-jenkins gap-jenkins \Chapter{The catalog of almost crystallographic groups}
This chapter introduces the access functions to the catalog of
3- and 4-dimensional crystallographic groups. This catalog is an
electronic version of the classification obtained in \cite{KD}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Rational matrix groups}
The following three main functions are available to access the library
of almost crystallographic groups as rational matrix groups.
\> AlmostCrystallographicGroup( , , )
\> AlmostCrystallographicDim3( , )
\> AlmostCrystallographicDim4( , )
is the dimension of the required group. Thus must be
either 3 or 4. The inputs and are used to define
the desired group as described in \cite{KD}. We outline the possible
choices for and here briefly. A more extended
description is given later in Section "More about almost crystallographic
groups" or can be obtained from \cite{KD}.
specifies the type of the required group. There are 17 types
in dimension 3 and 95 types in dimension 4. The input can either
be an integer defining the position of the desired type among all types;
that is, in this case is a number in [1..17] in dimension 3 or a
number in [1..95] in dimension 4. Alternatively, can be a string
defining the desired type. In dimension 3 the possible strings are
`"01"', `"02"', $\ldots$, `"17"'. In dimension 4 the possible strings
are listed in the list `ACDim4Types' and thus can be accessed from \GAP.
is a list of integers. Its length depends on the type of
the chosen group. The lists `ACDim3Param' and `ACDim4Param' contain
at position $i$ the length of the parameter list for the type number $i$.
Every list of integers of this length is a valid input.
Alternatively, one can input `false' instead of a parameter list. Then
\GAP\ will chose a random parameter list of suitable length.
\beginexample
gap> G := AlmostCrystallographicGroup( 4, 50, [ 1, -4, 1, 2 ] );
gap> DimensionOfMatrixGroup( G );
5
gap> FieldOfMatrixGroup( G );
Rationals
gap> GeneratorsOfGroup( G );
[ [ [ 1, 0, -1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, 1/2, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 1 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, -4, 1, 0, 1/2 ], [ 0, 0, -1, 0, 0 ], [ 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 1, 1/4 ], [ 0, 0, 0, 0, 1 ] ] ]
gap> G.1;
[ [ 1, 0, -1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ]
gap> ACDim4Types[50];
"076"
gap> ACDim4Param[50];
4
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Polycyclically presented groups}
All the almost crystallographic groups considered in this package are
polycyclic. Hence they have a polycyclic presentation and this can be
used to facilitate efficient computations with the groups. To obtain the
polycyclic presentation of an almost crystallographic group we supply the
following functions. Note that the share package {\sf Polycyclic} must be
installed to use these functions.
\> AlmostCrystallographicPcpGroup( , , )
\> AlmostCrystallographicPcpDim3( , )
\> AlmostCrystallographicPcpDim4( , )
The input is the same as for the corresponding matrix group functions.
The output is a pcp group isomorphic to the corresponding matrix group.
An explicit isomorphism from an almost crystallographic matrix group
to the corresponding pcp group can be obtained by the following function.
\> IsomorphismPcpGroup( )
We can use the polycyclic presentations of almost crystallographic
groups to exhibit structure information on these groups. For example,
we can determine their Fitting subgroup and ask group-theoretic
questions about this nilpotent group. The factor $G / Fit(G)$ of an
almost crystallographic group $G$ is called *holonomy group*. We
provide access to this factor of a pcp group via the following
functions. Let $G$ be an almost crystallographic pcp group.
\> HolonomyGroup( )
\> NaturalHomomorphismOnHolonomyGroup( )
The following example shows applications of these functions.
\beginexample
gap> G := AlmostCrystallographicPcpGroup( 4, 50, [ 1, -4, 1, 2 ] );
Pcp-group with orders [ 4, 0, 0, 0, 0 ]
gap> Cgs(G);
[ g1, g2, g3, g4, g5 ]
gap> F := FittingSubgroup( G );
Pcp-group with orders [ 0, 0, 0, 0 ]
gap> Centre(F);
Pcp-group with orders [ 0, 0 ]
gap> LowerCentralSeries(F);
[ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0 ],
Pcp-group with orders [ ] ]
gap> UpperCentralSeries(F);
[ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0, 0 ],
Pcp-group with orders [ ] ]
gap> MinimalGeneratingSet(F);
[ g2, g3, g4 ]
gap> H := HolonomyGroup( G );
Pcp-group with orders [ 4 ]
gap> hom := NaturalHomomorphismOnHolonomyGroup( G );
[ g1, g2, g3, g4, g5 ] -> [ g1, identity, identity, identity, identity ]
gap> U := Subgroup( H, [Pcp(H)[1]^2] );
Pcp-group with orders [ 2 ]
gap> PreImage( hom, U );
Pcp-group with orders [ 2, 0, 0, 0, 0 ]
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{More about the type and the defining parameters}
Each group from this library knows that it is almost crystallographic and,
additionally, it knows its type and defining parameters.
\> AlmostCrystallographicInfo( )
This attribute is set for groups from the library only. It is not possible
at current to determine the type and the defining parameters for an arbitrary
almost crystallographic groups which is not defined by the library access
functions.
\beginexample
gap> G := AlmostCrystallographicGroup( 4, 70, false );
gap> IsAlmostCrystallographic(G);
true
gap> AlmostCrystallographicInfo(G);
rec( dim := 4, type := 70, param := [ 1, -4, 1, 2, -3 ] )
\endexample
\beginexample
gap> G := AlmostCrystallographicPcpGroup( 4, 70, false );
Pcp-group with orders [ 6, 0, 0, 0, 0 ]
gap> IsAlmostCrystallographic(G);
true
gap> AlmostCrystallographicInfo(G);
rec( dim := 4, type := 70, param := [ -3, 2, 5, 1, 0 ] )
\endexample
We consider the types of almost crystallographic groups in more detail. The
almost crystallographic groups in dimensions 3 and 4 fall into three families
\beginlist
\item{(1)} 3-dimensional almost crystallographic groups.
\item{(2)} 4-dimensional almost crystallographic groups with a
Fitting subgroup of class 2.
\item{(3)} 4-dimensional almost crystallographic groups with a
Fitting subgroup of class 3.
\endlist
These families are split up further into subfamilies in \cite{KD} and to
each subfamily is assigned a type; that is, a string which is used to
identify the subfamily. As mentioned above, for the 3-dimensional almost
crystallographic groups the type is a string representing the numbers from
1 to 17, i.e. the available types are `"01"', `"02"', $\ldots$, `"17"'.
For the 4-dimensional almost crystallographic groups with a Fitting subgroup
of class 2 the type is a string of 3 or 4 characters. In general, a string of
3 characters representing
the number of the table entry in \cite{KD} is used. So possible types are
`"001"', `"002"', $\ldots$. The reader is warned however that not all
possible numbers are used, e.g.\ there are no groups of type `"016"'. Also,
the types do not appear in their natural order in \cite{KD}. Moreover, for
certain numbers there is more than one family of groups listed in \cite{KD}.
For example, the 3 families of groups corresponding to number 19 on pages
179-180 of \cite{KD} have types `"019"', `"019b"' and `"019c"' (the order is
the one given in \cite{KD}).
For the last category of groups, the 4-dimensional almost crystallographic
groups with a Fitting subgroup of class 3, the type is a string of 2 or 3
characters, where the first character is always the letter `"B"'. This `"B"'
is followed by the number of the table entry as found in \cite{KD},
eventually followed by a `"b"' or `"c"' as in the previous case.
For each type of almost crystallographic group contained in the library
there exists a function taking a parameter list as input and returning
the desired matrix or pcp group. These functions can be accessed
from \GAP\ using the lists `ACDim3Funcs', `ACDim4Funcs', `ACPcpDim3Funcs'
and `ACPcpDim4Funcs' which consist of the corresponding functions.
Although we include these direct access functions here for completeness,
we note that the user should in general use the higher-level functions
introduced above to obtain almost crystallographic groups from the
library. In particular, these low-level access functions return matrix
or pcp groups, but the almost crystallographic info flags will not be
attached to them.
\beginexample
gap> ACDim3Funcs[15];
function( k1, k2, k3, k4 ) ... end
gap> ACDim3Funcs[15](1,1,1,1);
gap> ACPcpDim3Funcs[1](1);
Pcp-group with orders [ 0, 0, 0 ]
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{The electronic versus the printed library}
The package `aclib' can be considered as the electronic version of
Chapter 7 of \cite{KD}. In this section we outline the relationship
between the library presented in this manual and the printed version
in \cite{KD}. First we consider an example. At page 175 of \cite{KD},
we find the following groups in the table starting with entry ``13''.
\medskip
13. $Q=P2/c$
$$
\matrix{ E:\;\langle a,b,c,d,\alpha,\beta\;\|\; & {\,[b,a]=1}\hskip 1.61cm
{[d,a]=1}
\hfill & \rangle \cr
& \matrix{ {[c,a]=d^{2 k_1}}\hfill & {[d,b]=1}\hfill \cr
{[c,b]=1}\hfill & {[d,c]=1}\hfill \cr
\alpha a=a^{-1}\alpha d^{k_{2}}\hfill & \alpha^2=d^{k_3}\hfill \cr
\alpha b=b\alpha \hfill & \alpha d= d \alpha\hfill \cr
\alpha c=c^{-1}\alpha d^{-2 k_6}\hfill & \cr
\beta a=a^{-1}\beta d^{k_1+k_2} \hfill & \beta^2=d^{k_5}\hfill \cr
\beta b=b^{-1}\beta d^{k_4}\hfill & \beta d= d \beta\hfill \cr
\beta c=c^{-1}\beta d^{-2 k_6}\hfill & \alpha \beta=c\beta\alpha d^{k_6}
\hfill } & }
$$
$$\lambda(\alpha)=\left(\matrix{
1& \frac{k_1}{2}+k_2 & 0 & -2 k_6 & \frac{k_3}{2}+\frac{k_6}{2} \cr
0& -1 & 0 & 0 & 0 \cr
0 & 0& 1 & 0 & 0 \cr
0 & 0 & 0 & -1 & \frac12\cr
0 & 0 & 0 & 0 & 1
}\right)
\;\;\lambda(\beta)=\left(\matrix{
1& k_1+k_2 & k_4 & -2 k_6 & \frac{k_5}{2} \cr
0& -1 & 0 & 0 & 0 \cr
0 & 0& -1 & 0 & 0 \cr
0 & 0 & 0 & -1 & 0\cr
0 & 0 & 0 & 0 & 1
}\right)
$$
$$H^2(Q,\Z{})=\Z{}\oplus(\Z{}_2)^4=\Z^{6}/A,$$
$$A=\{(k_1,\ldots,k_6)\|
k_1=0,\;k_2,\ldots, k_5\in 2\Z{},\;k_6\in\Z\}$$
AB-groups:
$\forall k>0,\;k\equiv 0\bmod 2,\;<(k,0,1,0,1,0)>$
\medskip
The number ``13'' at the beginning of this entry is the type of the
almost crystallographic group in this library. This family of groups
with type 13 depends on 6 parameters $k_1, k_2, \ldots, k_6$ and these
are the list in this library. The rational matrix
representation in \GAP\ corresponds exactly to the printed version in
\cite{KD} where it is named $\lambda$. In the example below, we consider
the group with parameters $(k_1,k_2,k_3,k_4,k_5,k_6)=(8,0,1,0,1,0)$.
\beginexample
gap> G:=AlmostCrystallographicDim4("013",[8,0,1,0,1,0]);
gap> G.5;
[ [ 1, 4, 0, 0, 1/2 ], [ 0, -1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, -1, 1/2 ], [ 0, 0, 0, 0, 1 ] ]
gap> G.6;
[ [ 1, 8, 0, 0, 1/2 ], [ 0, -1, 0, 0, 0 ], [ 0, 0, -1, 0, 0 ],
[ 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 1 ] ]
\endexample
For a 4-dimensional almost crystallographic group the matrix group is
built up such that $\{ a, b, c, d, \alpha, \beta, \gamma \}$ as described
in \cite{KD} forms the defining generating set of $G$. For certain types
the elements $\alpha$, $\beta$ or $\gamma$ may not be present.
Similarly, for a 3-dimensional group we have the generating set $\{ a, b,
c, \alpha, \beta \}$ and $\alpha$ and $\beta$ may be absent.
\bigskip
To obtain a polycyclic generating sequence from the defining generators
of the matrix group we have to order the elements in the generating set
suitably. For this purpose we take the subsequence of $(\gamma, \beta,
\alpha, a, b, c, d)$ of those generators which are present in the
defining generating set of the matrix group. This new ordering of the
generators is then used to define a polycyclic presentation of the given
almost crystallographic group.
aclib-1.3.2/tst/manual.example-3.tst 0000664 0003717 0003717 00000005432 13627151544 020432 0 ustar gap-jenkins gap-jenkins gap> START_TEST("");
gap> G := AlmostCrystallographicGroup( 4, 50, [ 1, -4, 1, 2 ] );
gap> DimensionOfMatrixGroup( G );
5
gap> FieldOfMatrixGroup( G );
Rationals
gap> GeneratorsOfGroup( G );
[ [ [ 1, 0, -1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, 1/2, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 1 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
[ [ 1, -4, 1, 0, 1/2 ], [ 0, 0, -1, 0, 0 ], [ 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 1, 1/4 ], [ 0, 0, 0, 0, 1 ] ] ]
gap> G.1;
[ [ 1, 0, -1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ]
gap> ACDim4Types[50];
"076"
gap> ACDim4Param[50];
4
gap> G := AlmostCrystallographicPcpGroup( 4, 50, [ 1, -4, 1, 2 ] );
Pcp-group with orders [ 4, 0, 0, 0, 0 ]
gap> Cgs(G);
[ g1, g2, g3, g4, g5 ]
gap> F := FittingSubgroup( G );
Pcp-group with orders [ 0, 0, 0, 0 ]
gap> Centre(F);
Pcp-group with orders [ 0, 0 ]
gap> LowerCentralSeries(F);
[ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0 ],
Pcp-group with orders [ ] ]
gap> UpperCentralSeries(F);
[ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0, 0 ],
Pcp-group with orders [ ] ]
gap> MinimalGeneratingSet(F);
[ g2, g3, g4 ]
gap> H := HolonomyGroup( G );
Pcp-group with orders [ 4 ]
gap> hom := NaturalHomomorphismOnHolonomyGroup( G );
[ g1, g2, g3, g4, g5 ] -> [ g1, id, id, id, id ]
gap> U := Subgroup( H, [Pcp(H)[1]^2] );
Pcp-group with orders [ 2 ]
gap> PreImage( hom, U );
Pcp-group with orders [ 2, 0, 0, 0, 0 ]
gap> G := AlmostCrystallographicGroup( 4, 70, false );
gap> IsAlmostCrystallographic(G);
true
gap> AlmostCrystallographicInfo(G);
rec( dim := 4, param := [ -3, 2, -1, -2, -1 ], type := 70 )
gap> G := AlmostCrystallographicPcpGroup( 4, 70, false );
Pcp-group with orders [ 6, 0, 0, 0, 0 ]
gap> IsAlmostCrystallographic(G);
true
gap> AlmostCrystallographicInfo(G);
rec( dim := 4, param := [ -1, 1, -4, 1, 0 ], type := 70 )
gap> ACDim3Funcs[15];
function( k1, k2, k3, k4 ) ... end
gap> ACDim3Funcs[15](1,1,1,1);
gap> ACPcpDim3Funcs[1](1);
Pcp-group with orders [ 0, 0, 0 ]
gap> G:=AlmostCrystallographicDim4("013",[8,0,1,0,1,0]);
gap> G.5;
[ [ 1, 4, 0, 0, 1/2 ], [ 0, -1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, -1, 1/2 ], [ 0, 0, 0, 0, 1 ] ]
gap> G.6;
[ [ 1, 8, 0, 0, 1/2 ], [ 0, -1, 0, 0, 0 ], [ 0, 0, -1, 0, 0 ],
[ 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 1 ] ]
#
gap> STOP_TEST( "" ,1);
aclib-1.3.2/tst/manual.example-4.tst 0000664 0003717 0003717 00000005053 13627151544 020432 0 ustar gap-jenkins gap-jenkins gap> START_TEST("");
gap> G:=AlmostCrystallographicPcpDim4("013",[8,0,1,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
true
gap> G:=AlmostCrystallographicPcpDim4("013",[9,0,1,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
false
gap> G:=AlmostCrystallographicPcpDim4("013",[10,0,2,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
false
gap> G:=AlmostCrystallographicPcpDim4("013",[10,0,3,0,1,9]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
true
gap> G:=AlmostCrystallographicPcpGroup(4, "085", false);
Pcp-group with orders [ 2, 4, 0, 0, 0, 0 ]
gap> GroupGeneratedByd:=Subgroup(G, [G.6] );
Pcp-group with orders [ 0 ]
gap> Q:=G/GroupGeneratedByd;
Pcp-group with orders [ 2, 4, 0, 0, 0 ]
gap> action:=List( Pcp(Q), x -> [[1]] );
[ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ]
gap> C:=CRRecordByMats( Q, action);;
gap> TwoCohomologyCR( C ).factor.rels;
[ 2, 2, 4, 0 ]
gap> G := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,0] );
Pcp-group with orders [ 2, 6, 0, 0, 0 ]
gap> projection := NaturalHomomorphismOnHolonomyGroup( G );
[ g1, g2, g3, g4, g5 ] -> [ g1, g2, id, id, id ]
gap> F := HolonomyGroup( G );
Pcp-group with orders [ 2, 6 ]
gap> IPprimeD6 := Subgroup( F , [F.2^2] );
Pcp-group with orders [ 3 ]
gap> K := PreImage( projection, IPprimeD6 );
Pcp-group with orders [ 3, 0, 0, 0 ]
gap> PrintPcpPresentation( K );
g1^3 = id
g2 ^ g1 = g2^-1 * g3^-1
g3 ^ g1 = g2 * g4^2
g3 ^ g2 = g3 * g4^2
g3 ^ g2^-1 = g3 * g4^-2
gap> Gamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));
Pcp-group with orders [ 0, 0, 0 ]
gap> quotient := G/Gamma3K;
Pcp-group with orders [ 2, 6, 3, 3, 2 ]
gap> S := SylowSubgroup( quotient, 3);
Pcp-group with orders [ 3, 3, 3 ]
gap> N := NormalClosure( quotient, S);
Pcp-group with orders [ 3, 3, 3 ]
gap> localization := quotient/N;
Pcp-group with orders [ 2, 2, 2 ]
gap> PrintPcpPresentation( localization );
g1^2 = id
g2^2 = id
g3^2 = id
gap> G := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,1]);;
gap> projection := NaturalHomomorphismOnHolonomyGroup( G );;
gap> F := HolonomyGroup( G );;
gap> IPprimeD6 := Subgroup( F , [F.2^2] );;
gap> K := PreImage( projection, IPprimeD6 );;
gap> Gamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));;
gap> quotient := G/Gamma3K;;
gap> S := SylowSubgroup( quotient, 3);;
gap> N := NormalClosure( quotient, S);;
gap> localization := quotient/N;
Pcp-group with orders [ 2, 2, 2 ]
gap> PrintPcpPresentation( localization );
g1^2 = id
g2^2 = g3
g3^2 = id
g2 ^ g1 = g2 * g3
#
gap> STOP_TEST( "" ,1);
aclib-1.3.2/tst/testall.g 0000664 0003717 0003717 00000000171 13627151544 016442 0 ustar gap-jenkins gap-jenkins LoadPackage("aclib");
TestDirectory(DirectoriesPackageLibrary("aclib", "tst"), rec(exitGAP := true));
FORCE_QUIT_GAP(1);
aclib-1.3.2/doc/manual.six 0000664 0003717 0003717 00000002227 13627151544 016563 0 ustar gap-jenkins gap-jenkins C intro.tex 1. The Almost Crystallographic Groups Package
S 1.1. More about almost crystallographic groups
C algos.tex 2. Algorithms for almost crystallographic groups
S 2.1. Properties of almost crystallographic groups
F 2.1. IsAlmostCrystallographic
F 2.1. IsAlmostBieberbachGroup
S 2.2. Betti numbers
F 2.2. OrientationModule
F 2.2. BettiNumber
F 2.2. BettiNumbers
S 2.3. Determination of certain extensions
F 2.3. HasExtensionOfType
C aclib.tex 3. The catalog of almost crystallographic groups
S 3.1. Rational matrix groups
F 3.1. AlmostCrystallographicGroup
F 3.1. AlmostCrystallographicDim3
F 3.1. AlmostCrystallographicDim4
S 3.2. Polycyclically presented groups
F 3.2. AlmostCrystallographicPcpGroup
F 3.2. AlmostCrystallographicPcpDim3
F 3.2. AlmostCrystallographicPcpDim4
F 3.2. IsomorphismPcpGroup
F 3.2. HolonomyGroup
F 3.2. NaturalHomomorphismOnHolonomyGroup
S 3.3. More about the type and the defining parameters
F 3.3. AlmostCrystallographicInfo
S 3.4. The electronic versus the printed library
C examp.tex 4. Example computations with almost crystallographic groups
S 4.1. Example computations I
S 4.2. Example computations II
S 4.3. Example computations III
aclib-1.3.2/doc/make_doc 0000775 0003717 0003717 00000001350 13627151544 016245 0 ustar gap-jenkins gap-jenkins #!/bin/sh
set -e
DEFAULT_GAPPATH=../../..
if test -z "$1"; then
GAPPATH=${DEFAULT_GAPPATH}; echo "Using ${DEFAULT_GAPPATH} as default GAP path";
else
GAPPATH=$1;
fi
echo "TeXing documentation"
# TeX the manual and build its bibliography
tex manual; bibtex manual
# TeX the manual again to incorporate the ToC ... and build the index
tex manual; $GAPPATH/doc/manualindex manual
# Finally TeX the manual again to get cross-references right
tex manual
# Create the .pdf version
pdftex manual; pdftex manual
# The HTML version of the manual
rm -rf ../htm
mkdir ../htm
echo "Creating HTML documentation"
$GAPPATH/etc/convert.pl -i -u -c -n aclib . ../htm
#############################################################################
##
#E
aclib-1.3.2/doc/manual.dvi 0000664 0003717 0003717 00000133424 13627151544 016546 0 ustar gap-jenkins gap-jenkins ; TeX output 2020.01.28:1612 p )& YNe9V $
phvb7tNAcGlibJvZ |X YA
GAP4PacGkag\%eOW 'Pe9V
phvb7tPComputations&with^ Almost&Cr!FystallographicGruoupsJ fbuyJ _ rKarel&Dekimpe(KULeuvenK.ulak) [ ܢand m0Bettina&Eicuk(Braunschweig) * p Լ\8PNContentsN Hߌ
ptmb7tH1$ The AlmostCrystallographicGroupsPgackage 3 ;3{
ptmr7t1.1$ More aboutalmostcrystallographicgroupsH . . . . . . . . . . . . . . . . . . . 3 H2$ Algorithms f oralmostcrystallographicgroups 5 2.1$ Properties ofalmostcrystallographicgroupsY . . . . . . . . . . . . . . . . . . . 5 2.2$ Betti numbers\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3$ Determination ofcertaine٠xtensions
) . . . . . . . . . . . . . . . . . . . . . 6 H3$ The catalogofalmostcrystallographicgroups m7 3.1$ Rational matrixgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2$ Polyc٠yclically presentedgroupsd. . . . . . . . . . . . . . . . . . . . . . . . 8 3.3$ More aboutthetypeandthede ningparameters / . . . . . . . . . . . . . . . . . 9 3.4$ The electronicv٠ersustheprintedlibraryh . . . . . . . . . . . . . . . . . . . . 10 H4$ Example computationswithalmostcrystallographicgroups 712 4.1$ Example computationsI
;& . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2$ Example computationsII殍 . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3$ Example computationsIII6 . . . . . . . . . . . . . . . . . . . . . . . . . 13$ HBibliograph٠ynC16$ Index17 p ƢMe9V `
phvb7tM1
ENThe Almost, >Cr\%ystallographic GrGoups Packag\%e<ٍA]group]is]calledHalmostcrystallographicifitisa nitelygeneratednilpotent-by- nitegroupwithoutnon-tri vial nitenormalsubgroups.AnimportantspecialcaseofalmostcrystallographicgroupsaretheHalmostBieberbachgroups: thesearealmostcrystallographicandtorsionfree.N8Byitsde nition,analmostcrystallographicgroupLKj
ptmri7tLGhasa nitelygeneratednilpotentnormalsubgroupLNDof niteinde٠x.ClearlyY,LNN#ispolycyclicandthushasapolyc٠yclicseries.Thenumberofin nitec٠yclicfgactorsinsuchaseriesfor LNisaninv ariantofLG:theHHirschlengthofLG.F٠or#V
zptmcm7t3ӆDƛ
zptmcm7yD3{matriceswithdrealentriesisofthistype.ThenLL-oӆAut}(LL)actsanely(onthe left)onLLvia YGD8LlA?
zptmcm7mA;LlEƛ
zptmcm7yE06lD2LLA;D8A D2Aut(LL):0(TKj
ptmri7tTlB?
zptmcm7mB; )`}LlE06l=Ll0A x(LlE0oT)D 4 lLChapter 1.TheAlmostCrystalloggr٠aphicGroupsP37ackagep LetBLCzVbeamaximalcompactsubgroupofBAutA(LL).ThenasubgroupLGofLLZoSULCissaidtobeanalmostcrystallographic group\]ifandonlyiftheactionofLG\^onLL,inducedbytheactionofLLoAut(LL),isproperlydiscontinuousandthequotientspaceLGDnLLiscompact.Onereco٠versthesituationoftheordinarycrystallographicgroupsbytakingLL=R^Tn ,for someLn,andLCh=LO(Ln),theorthogonalgroup.N8More;generallyY,wesaythatanabstractgroupisanalmostcrystallographicgroupifitcanberealizedasagenuinealmostcrystallographicsubgroupofsomeLL o (LC8P.Inthefollo wingtheoremweoutlinesomealgebraiccharacterizationsofFalmostcrystallographicGgroups;seeTheorem3.1.3of[Dek96].RecallthattheHFittingsubgroupFitt(LG)ofapolyc٠yclic-by- nite groupLGisitsuniquemaximalnormalnilpotentsubgroup.HTheorem. Thefollo wingareequivalentforapolyc٠yclic-by- nitegroupLG:N8W(1) LG isanalmostcrystallographicgroup.W(2) Fitt(LG) istorsionfreeandof niteinde٠xinLG.W(3) LG containsatorsionfreenilpotentnormalsubgroupLNof niteinde٠xinLGwithLCTG(LN )torsionfree.W(4) LG hasanilpotentsubgroupof niteinde٠xandtherearenonon-tri vial nitenormalsubgroupsinLG.pIn particular,ifLGisalmostcrystallographic,thenLGA=LFitt.(LG)is nite.ThisfgactoriscalledtheHholonomygroupofLG.TheJdimensionIofanalmostcrystallographicgroupequalsthedimensionoftheLiegroupLL abo٠vewhichJcoincidesalsoˆwith˅theHirschlengthofthepolyc٠yclic-by- nitegroup.Thislibrarythereforecontainsfgamiliesofvirtuallynilpotent groupsofHirschlength3and4. ) p CM2 ^NNAlgorithms
2fGoralmost, !cr\%ystallographicZqgrGoups;Thisjchapteripresentsav arietyofalgorithmsforalmostcrystallographicgroups.Inmostcases,the٠yassumeapoly- c٠yclicallypresentedgroupasinput;inparticular,theinputgroupsmustbepolycyclicinthiscase.ThemethodsdescribedYhereXsupplementthemethodsoftheQPolycyclicpackageforpolyc٠yclicallypresentedgroups.Man٠yofthefunctions7in6thischapterarebasedonmethodsoftheQPolycyclicpackageandthusthispackagemustbeinstalledtousetheWIfunctionsintroducedWHhere.W37erefertotheQPolycyclicpackageforfurtherinformationonpolyc٠yclicpresentations. N8P2.1 PruoperBties&ofalmostcr!Fystallographicgroups 3S3{
ptmr7tS1̟^3u7 msam7IIsAlmostCrystallographic(?LG)48PN8Thisfunctionchecksifapolyc٠yclicallypresentedgroupLGxisalmostcrystallographic;thatis,itchecksifLGxisnilpotent-by- nite andhasnonon-tri vial nitenormalsubgroup.3S2̟^IIsAlmostBieberbachGroup(?LG)9xPN8This function checksifapolyc٠yclicallypresentedgroupLGisalmostBieberbach;thatis,itchecksifLGisnilpotent-by- nite andtorsionfree. N8P2.2 Betti&number sN8Let- LG- beapolyc٠yclicallypresentedandtorsionfreegroupofHirschlengthLn.ThenwecancomputetheBettinumbersA Tir((LG)pforpLiD2f0A;1A;2A;Ln D 2A;Ln D 1A;LnDg.pIfpLnD6,thenwepcancomputeallBettinumbersA Tir((LG)for0DLiD6ofpLG.W37eintroduce.thefollo wingfunctionsfor/thispurposeandwereferto[Bro82]forthedetailsontheorientationmoduleand theBettinumbers.3S1̟^IOrientationModule(?LG)XFN8This function determinestheorientationmoduleofthepolyc٠yclicallypresentedgroupLG;thatis,itreturnsalistofmatrices Lm1 A;:::;LmTnDLGL(1A;Z)whicharetheimagesofthe'Igs(G)'intheiractionontheorientationmodule.3S2̟^IBettiNumber(?LG,Lm)f;FN8ThisIfunctionreturnstheJLmthBettinumberofthepolyc٠yclicallypresentedtorsionfreegroupLG-ifLmSD2f0A;1A;2A;LnD 2A;Ln8D 1A;LnDg, whereLnistheHirschlengthofLG.3S3̟^IBettiNumbers(?LG)qAN8This.function/returnstheBettinumberofthepolyc٠yclicallypresentedtorsionfreegroupLG iftheHirschlengthofLGis smallerthan7. ' 6 LChapter 2.Algorithmsforalmostcrystalloggr٠aphicgroupsp P2.3 Determination&ofcerBtaine xtensionsN8LetaLGbbeapolyc٠yclicallypresentedalmostcrystallographicgroup.W37ewganttocheckthee٠xistenceofcertaine٠xten- sions ofLG.N8First,(mitiswell-kno wnthat(ltheequivalenceclassesofe٠xtensionsofLGcorrespond(ltothesecondcohomologygroupof LG.Thiscohomologygroupcanbecomputedusingthe methodsoftheQPolycyclicpackageforan٠yexplicitlygi venmoduleofLG.Further,wecanconstructapolyc٠yclicpresentationforeachcocycleofthesecondcohomologygroup.W37e gi v٠eanexampleforsuchacomputationbelo wY.Ho wev٠er,wemaybeinterestedincertaine٠xtensionsonly;fore٠xample,thetorsionfreee٠xtensionsareoftenofparticu-larW@interest.IfthesecondW?cohomologygroupis nite,thenwecancomputeapolyc٠yclicpresentationforeachelementofathisgroupandchecktheresultinggroupfortorsion`freeness.Butifthesecondcohomologygroupisin nite,thenthisLapproachisnotav ailable.HenceweintroducetheLfollowingspecialmethodtoco٠verLthisandrelatedapplications. 3S1̟^IHasExtensionOfType(?LG,LtorgsionfrGee,Lminimalcentre) FN8SupposeCthatLGC|isapolyc٠yclicallypresentedalmostcrystallographicgroupCwithFittingsubgroupLN .ThisfunctionchecksifthereisaLG-moduleLMW"D("=PZwhichiscentralizedbyLNsuchthattheree٠xistsatorsionfreee٠xtensionofLMbyLGg(ifthe
agLtorgsionfrGeeistrue)orane٠xtensionLE-dwithLZ (LFitt.(LE:))(=LMX(ifthe
agLminimalcentreistrue)orane٠xtension whichsatis esbothconditions(ifboth
agsaretrue).W37enotethatthee٠xistenceofsuche٠xtensionsisofinterestinthedeterminationofe٠xtensionswhicharealmostBieberbach groups. W37ereferto[DE01b]foramoredetailedaccountofthisapplicationandforfurtherresultsofasimilar nature. 3O p CM3 a9NThe
=catalog
=ofalmost, !cr\%ystallographicZqgrGoups;*Thischapterintroducestheaccessfunctionstothecatalogof3-and4-dimensionalcrystallographicgroups.This catalog isanelectronicv٠ersionoftheclassi cationobtainedin[Dek96]. ZP3.1 Rational&matrixgruoups-The0?follo wingthreemainfunctionsareavailableto0@accessthelibraryofalmostcrystallographicgroupsasrationalmatrix groups.q3S1̟^IAlmostCrystallographicGroup(?Ldim,Ltype,Lpar٠ametergs)36IAlmostCrystallographicDim3(?Ltype,Lpar٠ametergs)36IAlmostCrystallographicDim4(?Ltype,Lpar٠ametergs)5Ldimisthedimensionoftherequiredgroup.ThusLdimmustbeeither3or4.TheinputsLtypeandLpar٠ametergsareusedtoU de nethedesiredgroupasdescribedin[Dek96].W37eoutlinethepossiblechoicesforLtypeandLpar٠ametergsherebrie
yY. Amoree٠xtendeddescriptionisgi venlaterinSection1.1orcanbeobtainedfrom[Dek96].=yLtypeDspeci esthetypeDoftherequiredgroup.Thereare17typesindimension3and95typesindimension4.TheinputlLtypelcaneitherbeaninte٠gerde ningthepositionofthedesiredtypeamongalltypes;thatis,inthiscaseLtypeisanumberin[1..17]indimension3oranumberin[1..95]indimension4.Alternati v٠elyY,Ltypecanbeastringde ningthedesiredtype.Indimension3thepossiblestringsare"01","02",A:::|,"17".Indimension4thepossiblestringsare listedinthelistACDim4TypesandthuscanbeaccessedfromQGAP.=xLpar٠ametergsxisalistofyintegers.Itslengthdependsonthetypeofthechosenygroup.ThelistsACDim3ParamandACDim4ParamAcontainBatpositionLithelengthoftheparameterlistforthetypenumberLi.Ev٠erylistofinte٠gersofthislength8cis8dav alidLpar٠ametersBinput.Alternativ٠elyY,8conecaninputfalseinsteadofaparameterlist.ThenQGAPwillchosea randomparameterlistofsuitablelength.9gap>?G:=AlmostCrystallographicGroup(4,50,[1,-4,1,2]);99gap>?DimensionOfMatrixGroup(G);959gap>?FieldOfMatrixGroup(G);9Rationals9gap>?GeneratorsOfGroup(G);9[?[[1,0,-1/2,0,0],[0,1,0,0,1],[0,0,1,0,0],-[?0,0,0,1,0],[0,0,0,0,1]],[?[1,1/2,0,0,0],[0,1,0,0,0],[0,0,1,0,1],-[?0,0,0,1,0],[0,0,0,0,1]],[?[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],-[?0,0,0,1,1],[0,0,0,0,1]],[?[1,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0], < 8 讥LChapter 3.Thecataloggofalmostcrystallogr٠aphicgroupsp -[?0,0,0,1,0],[0,0,0,0,1]], [?[1,-4,1,0,1/2],[0,0,-1,0,0],[0,1,0,0,0],-[?0,0,0,1,1/4],[0,0,0,0,1]]]9gap>?G.1;9[?[1,0,-1/2,0,0],[0,1,0,0,1],[0,0,1,0,0],[?0,0,0,1,0],[0,0,0,0,1]]9gap>?ACDim4Types[50];9"076"9gap>?ACDim4Param[50];94 P3.2 Pzol ycyculically&presentedgroupsEAll'the(almostcrystallographicgroupsconsideredinthispackagearepolyc٠yclic.Hencethe٠yhavea(polycyclicpresen-tationMandLthiscanbeusedtofgacilitateecientcomputationswiththegroups.T37oobtainthepolyc٠yclicpresentationofanalmostcrystallographicgroupwesupplythefollo wingfunctions.NotethatthesharepackageQPolycyclicmustbe installedtousethesefunctions.⍍3S1̟^IAlmostCrystallographicPcpGroup(?Ldim,Ltype,Lpar٠ametergs)36IAlmostCrystallographicPcpDim3(?Ltype,Lpar٠ametergs)36IAlmostCrystallographicPcpDim4(?Ltype,Lpar٠ametergs)GThe inputis thesameasforthecorrespondingmatrixgroupfunctions.Theoutputisapcpgroupisomorphictothecor-responding1Fmatrix1Ggroup.Ane٠xplicitisomorphismfromanalmostcrystallographicmatrixgrouptothecorrespondingpcp groupcanbeobtainedbythefollo wingfunction.ፍ3S2̟^IIsomorphismPcpGroup(?LG)GW37ecanusethepolyc٠yclicpresentationsofalmostcrystallographicgroupstoexhibitstructureinformationonthesegroups.F٠orexample,wecandeterminetheirFittingsubgroupandaskgroup-theoreticquestionsaboutthisnilpotentgroup.ѯThefgactorLGA=LFit.(LG)ofanalmostcrystallographicgroupѮLGiscalledHholonomygroup.W37epro٠videaccesstothis fgactorofapcpgroupviathefollo wingfunctions.LetLGbeanalmostcrystallographicpcpgroup.3S3̟^IHolonomyGroup(?LG)36INaturalHomomorphismOnHolonomyGroup(?LG)GThe follo winge٠xampleshowsapplicationsofthesefunctions.9gap>?G:=AlmostCrystallographicPcpGroup(4,50,[1,-4,1,2]);9Pcp-group?withorders[4,0,0,0,0]9gap>?Cgs(G);9[?g1,g2,g3,g4,g5] 9gap>?F:=FittingSubgroup(G);9Pcp-group?withorders[0,0,0,0]9gap>?Centre(F);9Pcp-group?withorders[0,0]9gap>?LowerCentralSeries(F);9[?Pcp-groupwithorders[0,0,0,0],Pcp-groupwithorders[0],Pcp-group?withorders[
]]9gap>?UpperCentralSeries(F);9[?Pcp-groupwithorders[0,0,0,0],Pcp-groupwithorders[0,0],Pcp-group?withorders[
]]9gap>?MinimalGeneratingSet(F);9[?g2,g3,g4] H| LSection 3.MorGeaboutthetypeandthede ningpar٠ametergs 29p 9gap>?H:=HolonomyGroup(G); 9Pcp-group?withorders[4]9gap>?hom:=NaturalHomomorphismOnHolonomyGroup(G);9[?g1,g2,g3,g4,g5]->[g1,identity,identity,identity,identity]9gap>?U:=Subgroup(H,[Pcp(H)[1]^2]);9Pcp-group?withorders[2]9gap>?PreImage(hom,U);9Pcp-group?withorders[2,0,0,0,0] P3.3 More&aboutthetypeandthede ningparameter sٍEachgroupfromthislibrarykno wsthatitisalmostcrystallographicand,additionallyY,itkno wsitstypeandde ningparameters.3S1̟^IAlmostCrystallographicInfo(?LG)WThis\attribute\issetforgroupsfromthelibraryonlyY.Itisnotpossibleatcurrenttodeterminethetypeandthede ningparameters foranarbitraryalmostcrystallographicgroupswhichisnotde nedbythelibraryaccessfunctions.9gap>?G:=AlmostCrystallographicGroup(4,70,false);99gap>?IsAlmostCrystallographic(G);9true9gap>?AlmostCrystallographicInfo(G);9rec(?dim:=4,type:=70,param:=[1,-4,1,2,-3])>
9gap>?G:=AlmostCrystallographicPcpGroup(4,70,false);9Pcp-group?withorders[6,0,0,0,0]9gap>?IsAlmostCrystallographic(G);9true9gap>?AlmostCrystallographicInfo(G);9rec(?dim:=4,type:=70,param:=[-3,2,5,1,0])ٍW37e`considerthetypesof`almostcrystallographicgroupsinmoredetail.Thealmostcrystallographicgroupsindimen-sions 3and4fgallintothreefamiliesWW(1) 3-dimensional almostcrystallographicgroups.ԍW(2) 4-dimensional almostcrystallographicgroupswithaFittingsubgroupofclass2.ՍW(3) 4-dimensional almostcrystallographicgroupswithaFittingsubgroupofclass3.+These(Mfgamiliesaresplitupfurther(Lintosubfamiliesin[Dek96]andtoeachsubfamilyisassigned(Latype;thatis,astringwhich is usedtoidentifythesubfgamilyY.Asmentionedabo٠ve,for the3-dimensionalalmostcrystallographicgroupsthe typeisastringrepresentingthenumbersfrom1to17,i.e.theav ailabletypesare"01","02",A:::$,"17".F٠or!the4-dimensional!almostcrystallographicgroupswithaFittingsubgroupofclass2thetypeisastringof3or4V4characters.V3Ingeneral,astringof3charactersrepresentingthenumberofthetableentryin[Dek96]isused.Sopossibletypesare"001","002",A:::d/.Thereaderiswgarnedho wev٠erthatnotallpossiblenumbersareused,e.g.therenarenongroupsoftype"016".Also,thetypesdonotappearintheirnaturalorderin[Dek96].Moreo٠ver,nforcertainv~numbersvthereismorethanonefgamilyofgroupslistedin[Dek96].F٠orexample,v~the3fgamiliesofgroupscorrespondingtonumber19onpages179-180of[Dek96]hav٠etypes"019","019b"and"019c"(theorderistheone gi v٠enin[Dek96]).ԍF٠orthelastcategoryofgroups,the4-dimensionalalmostcrystallographicgroupswithaFittingsubgroupofclass3,thetypeisastringof2or3characters,wherethe rstcharacterisalgwaystheletter"B".This"B"isfollo wedbythenumber ofthetableentryasfoundin[Dek96],e v٠entuallyfollowedbya"b"or"c"asinthepreviouscase.
Tʍ 10 㮥LChapter 3.Thecataloggofalmostcrystallogr٠aphicgroupsp F٠oreachtypeofalmostcrystallographicgroupcontainedinthelibrarytheree٠xistsafunctiontakingaparameter listzasinputandreturningthedesiredmatrixorpcpzgroup.ThesefunctionscanbeaccessedfromQGAPusingthelistsACDim3Funcs,ACDim4Funcs,ACPcpDim3FuncsandACPcpDim4Funcswhichconsistofthecorrespondingfunctions.N8Although.we-includethesedirectaccessfunctionshereforcompleteness,wenotethattheusershouldingeneralusethe-?higher-le v٠elfunctionsintroducedabovetoobtainalmostcrystallographicgroupsfromthelibraryY.Inparticular,theseQlo w-lev٠elQaccessfunctionsreturnmatrixorpcpgroups,butthealmostcrystallographicinfo
agswillnotbeattached tothem. 9gap>?ACDim3Funcs[15];9function(?k1,k2,k3,k4)...end9gap>?ACDim3Funcs[15](1,1,1,1);99gap>?ACPcpDim3Funcs[1](1);9Pcp-group?withorders[0,0,0] P3.4 The&electruonicver sustheprintedlibrar!FyN8The `packageaclibcan _beconsideredastheelectronicv٠ersionofChapter7of[Dek96].Inthissectionweoutlinethe}relationshipbetweenthe|librarypresentedinthismanualandtheprintedv٠ersionin[Dek96].Firstweconsiderane٠xample. Atpage175of[Dek96],we ndthefollo winggroupsinthetablestartingwithentry\13".13. LQ=LP2A=Lc: m1?E:0DhLaA;LbA;LcA;LdEA; x; Dj D[LbA;La]=1"aJ[LdEA;La]=1bDi5 D[LcA;La]=LdE^2Tk @V
zptmcm7t@1 [LdEA;Lb]=1 D[LcA;Lb]=1 [LdEA;Lc]=1 DA xLa=La^E 1 p]A LdE^Tk @2 A x^2=LdE^Tk @3 DA xLb=LbA xLd 0=LdEA D xLc=Lc^E 1 p]A LdE^E 2Tk @6 DA :La=La^E 1 p]A LdE^Tk @1 +Tk @2 A :^2=LdE^Tk @5 DA :Lb=Lb^E 1 p]A LdE^Tk @4 A :Ld 0=LdEA D :Lc=Lc^E 1 p]A LdE^E 2Tk @6 A x =LcA : LdE^Tk @6=8 3A:(A x)=hGCq
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10(=0G,d0b)0{1h qYG1 qYC qYC qYCcЍ qYA&8 LHs02s0(LQA;Z)=Z8D(Z2 )4=Z6A=LAA; LA=Df(Lk1 A;:::;Lk6)DjLk1=0A;qLk2A;:::;Lk5D22ZA;qLk6D2ZDgAB-groups:N8D8LkA>0A;qLkD0modU2A;q(Lk+A;L0A;L1A;L0A;L1A;L0)The
,number
+\13"atthebe٠ginningofthisentryisthetypeofthealmostcrystallographicgroupinthislibraryY.This fgamily of groupswithtype13dependson6parametersLk1 A;Lk2A;:::;Lk6 and these aretheLpar٠ametergslistinthislibraryY.TheThrationalmatrixrepresentationinQGAPcorrespondse٠xactlytotheprintedversionin[Dek96]whereitisnamedA:.In thee٠xamplebelo wY,weconsiderthegroupwithparameters(Lk1 A;Lk2A;Lk3A;Lk4A;Lk5A;Lk6)=(8A;0A;1A;0A;1A;0). c LSection 4.Theelectronicvergsustheprintedlibr٠ary 11p 9gap>?G:=AlmostCrystallographicDim4("013",[8,0,1,0,1,0]); 99gap>?G.5;9[?[1,4,0,0,1/2],[0,-1,0,0,0],[0,0,1,0,0],[?0,0,0,-1,1/2],[0,0,0,0,1]]9gap>?G.6;9[?[1,8,0,0,1/2],[0,-1,0,0,0],[0,0,-1,0,0],[?0,0,0,-1,0],[0,0,0,0,1]]N8F٠orBSa4-dimensionalalmostcrystallographicBRgroupthematrixgroupisbuiltupsuchthatDfLaA;LbA;LcA;LdEA; x; :;
DgBSasde-scribed8in[Dek96]formsthede ning9generatingsetofLG.F٠orcertaintypestheelementsA x,A orA
maynotbepresent. SimilarlyY,fora3-dimensionalgroupwehav٠ethegeneratingsetDfLaA;LbA;LcA; x; :Dg andA TxandA maybeabsent. N8T37o\obtain\apolyc٠yclicgeneratingsequencefromthede ninggeneratorsofthematrixgroupwehav٠etoordertheelementsHintheHgeneratingsetsuitablyY.F٠orthispurposewetakgethesubsequenceof(A
x; :; ;LaA;LbA;LcA;LdE)HofHthosegener-atorswhicharepresentinthede ninggeneratingsetofthematrixgroup.Thisne worderingofthegeneratorsisthenused tode neapolyc٠yclicpresentationofthegi venalmostcrystallographicgroup. v p YƢM4P ?NExample, acomputations 8with .almost !cr\%ystallographicZqgrGoups>ʍP4.1 Example&computationsIN8Usingthefunctionsav ailableforpcpgroupsinthesharepackageQpolycyclicitisnoweasytoredosomeofthe calculationswofw[Dek96].Asa rste٠xamplewecheckwhetherthegroupsindicatedastorsionfreein[Dek96]arealsodeterminedastorsionfreeonesbyQGAP.In[Dek96]thesealmostBieberbachgroupsarelistedas\37AB-groups".SoforJtypeJ\013"thesearethegroupswithparameters(Lk+A;0A;1A;0A;1A;0)JwhereLkvisane v٠enintegers8.JLet'slookJatsomee٠xamples inQGAP: 9gap>?G:=AlmostCrystallographicPcpDim4("013",[8,0,1,0,1,0]);9Pcp-group?withorders[2,2,0,0,0,0]9gap>?IsTorsionFree(G);9true9gap>?G:=AlmostCrystallographicPcpDim4("013",[9,0,1,0,1,0]);9Pcp-group?withorders[2,2,0,0,0,0]9gap>?IsTorsionFree(G);9falseFurther,thereisalsosomecohomologyinformationinthetablesof[Dek96].Infgact,thegroupsinthislibrarywereobtained ase٠xtensionsLEoftheformN8 Ɵ1D!ZD!LED!LQD!1N8where,inthe4-dimensionalcaseLQ=LE:A=DhLdEDi.Thecohomologyinformationfortheparticulare٠xampleabovesho wsthatthegroupsdeterminedbyaparameterset(Lk1 A;Lk2A;Lk3A;Lk4A;Lk4A;Lk6)areequi valentase٠xtensionstothegroupsdeter-mined bytheparameters(Lk1 A;Lk2modU2A;Lk3mod2A;Lk4mod2A;Lk5mod2A;0). Thisisalsovisiblein ndingtorsion:
| LSection 3.ExamplecomputationsIII8013p 9gap>?G:=AlmostCrystallographicPcpDim4("013",[10,0,2,0,1,0]); 9Pcp-group?withorders[2,2,0,0,0,0]9gap>?IsTorsionFree(G);9false9gap>?G:=AlmostCrystallographicPcpDim4("013",[10,0,3,0,1,9]);9Pcp-group?withorders[2,2,0,0,0,0]9gap>?IsTorsionFree(G);9true P4.2 Example&computationsIIN8The"computation"ofcohomologygroupsplayedanimportantroleintheclassi cationofthealmostBieberbachgroupsin[Dek96].UsingQGAP,itisno wpossibletocheckthesecomputations.Asane٠xampleweconsiderthe4-dimensionalalmostRcrystallographicQgroupsoftype85onpage202of[Dek96].ThisgroupLE10has6generators.Inthetable,onealso ndstheinformationN8 ]LHs02s0(LQA;Z)=Z8D(Z2 )28DZ4N8for LQ=LE:A=DhLdEDi asabo٠ve. Moreover,theLQ{module ZisinfgactthegroupDhLdEDi,wheretheLQ-actioncomesfromconjug7ationinsideLE:.Inthecaseofgroupsoftype85,Zisatri vialLQ-module.Thefollo winge٠xampledemonstratesho w to(re)computethistwgo-cohomologygroupLHs0^2s0(LQA;Z). 9gap>?G:=AlmostCrystallographicPcpGroup(4,"085",false);9Pcp?groupwithorders[2,4,0,0,0,0]9gap>?GroupGeneratedByd:=Subgroup(G,[G.6]);9Pcp?groupwithorders[0]9gap>?Q:=G/GroupGeneratedByd;9Pcp?groupwithorders[2,4,0,0,0]9gap>?action:=List(Pcp(Q),x->[[1]]);9[?[[1]],[[1]],[[1]],[[1]],[[1]]]9gap>?C:=CRRecordByMats(Q,action);;9gap>?TwoCohomologyCR(C).factor.rels;9[?2,2,4,0]ThisMlastlinegi v٠esMustheabelianinvariantsofthesecondMcohomologygroupLHs0^2s0(LQA;Z).SoweshouldreadthislineasN8 cLHs02s0(LQA;Z)=Z28D8Z2DZ4DZN8which indeedcoincideswiththeinformationin[Dek96]. N8P4.3 Example&computationsIIIAsanotherapplicationofthecapabilitiesofthecombinationofaclibandQpolycyclicwechecksomecomputationsof [DM01].N8Sectionay5ofazthepaper[DM01]iscompletelyde votedtoane٠xampleofthecomputationoftheLP-localizationofavirtuallyCnilpotentgroup,whereDLPisasetofprimes.Althoughitisnotourintentiontode v٠elopthetheoryofLP-localization ofgroupsatthisplace,letussummarizesomeofthemainresultsconcerningthistopichere.F٠orasetofprimesLP,wesaythatLnJDD2JELPifandonlyifLnisaproductofprimesinLP.AgroupLGissaidtobeLP-localifXandYonlyifthemapATn:LGD!LG:LgD7!Lg^TnYisXbijecti v٠eforallLnD2LP^E0l,whereXLP^E0$isthesetofallprimesnotinLP.TheZ#LP-localizationZ"ofagroupLG,isaLP-localgroupLGTP! togetherwithamorphismA /:[ LG[
D!LGTP! whichsatisfyZ#the 14 @LChapter 4.Examplecomputationswithalmostcrystalloggr٠aphicgroupsp follo wingFKuniv٠ersalFLproperty:ForFLeachLP-localgroupLLMandan٠ymorphismA':LGD!LL,thereFLe٠xistsauniquemorphism A :LGTPD!LL, suchthatA
XD8A =A' x.ThisÂconceptofÃlocalizationiswellde v٠elopedfor nitegroupsandfornilpotentgroups.F٠ora nitegroupLG,theLP-localizationisthelargestquotientofLG,havingnoelementswithanorderbelongingtoLP^E0V(themorphismA x,mentionedabo٠ve isthenaturalprojection).In[DM01]acontributionismadeto wgardsthelocalizationofvirtuallynilpotentgroups.Thetheorydev٠elopedinthegpaperisthenillustratedinthelastsectionofthefpaperbymeansofthecomputationoftheLP-localizationofan almostcrystallographicgroup. F٠ortheirexampletheauthorshavechosenan almostcrystallographicgroupLGofdimension3andtype17.F٠orthesetofparameters(Lk1 A;Lk2A;Lk3A;Lk4)the٠yhaveconsideredallcasesoftheform(Lk1 A;Lk2A;Lk3A;Lk4)=(2A;0A;0A;Lk4 ).HerewewillchecktheircomputationsintwgocasesLk4 =0andLk4=1usingthesetofprimesLP=Df2Dg.Theholonomy(groupofthesealmostcrystallographicgroupsLGisthedihedral)groupDD6(oforder12.Thusthereisashorte٠xact sequenceoftheform 1D!Fitte(LG)D!LGD!D6D!1Dz Asa rststepintheircomputation,DescheemaekgerandMalfgaitdeterminethegroupLITP"Fƛ
zptmcm7yF0DD6 ,whichistheuniquesubgroupyoforder3inDD6 .Oneofxthemainobjectsin[DM01]isthegroupLK =Lp^E 1 p](LITP"F0DD6),whereLpisthenaturalprojectionofLGontoitsholonomygroup.Itiskno wnthattheLP-localizationofLGcoincideswiththeLP-localizationofLGA=
3 (LK ),whereA
3(LK )isthethirdterminthelo wercentralseriesofLK.AsLGA=
3 (LK)is niteinthise٠xample,wee٠xactlykno wwhatthisLP-localizationis.Letusno wshowY,howGAP canbeusedtocomputethisLP-localizationintwgo cases:First case:Theparametersare(Lk1 A;Lk2A;Lk3A;Lk4)=(2A;0A;0A;0)9gap>?G:=AlmostCrystallographicPcpGroup(3,17,[2,0,0,0]);9Pcp?groupwithorders[2,6,0,0,0]9gap>?projection:=NaturalHomomorphismOnHolonomyGroup(G);9[?g1,g2,g3,g4,g5]->[g1,g2,identity,identity,identity]9gap>?F:=HolonomyGroup(G);9Pcp?groupwithorders[2,6]9gap>?IPprimeD6:=Subgroup(F,[F.2^2]);9Pcp?groupwithorders[3]9gap>?K:=PreImage(projection,IPprimeD6);9Pcp?groupwithorders[3,0,0,0]9gap>?PrintPcpPresentation(K);9pcp?presentationongenerators[g2^2,g3,g4,g5]9g2^2?^3=identity9g3?^g2^2=g3^-1*g4^-19g3?^g2^2^-1=g4*g5^-29g4?^g2^2=g3*g5^29g4?^g2^2^-1=g3^-1*g4^-1*g5^29g4?^g3=g4*g5^29g4?^g3^-1=g4*g5^-29gap>?Gamma3K:=CommutatorSubgroup(K,CommutatorSubgroup(K,K));9Pcp?groupwithorders[0,0,0]9gap>?quotient:=G/Gamma3K;9Pcp?groupwithorders[2,6,3,3,2]9gap>?S:=SylowSubgroup(quotient,3);9Pcp?groupwithorders[3,3,3]9gap>?N:=NormalClosure(quotient,S);9Pcp?groupwithorders[3,3,3]9gap>?localization:=quotient/N; ꍟ LSection 3.ExamplecomputationsIII8015p 9Pcp?groupwithorders[2,2,2] 9gap>?PrintPcpPresentation(localization);9pcp?presentationongenerators[g1,g2,g3]9g1?^2=identity9g2?^2=identity9g3?^2=identity This sho wsthatLGTPWD(=1Z^3d2 .N8Second case:Theparametersare(Lk1 A;Lk2A;Lk3A;Lk4)=(2A;0A;0A;1)9gap>?G:=AlmostCrystallographicPcpGroup(3,17,[2,0,0,1]);; 9gap>?projection:=NaturalHomomorphismOnHolonomyGroup(G);;9gap>?F:=HolonomyGroup(G);;9gap>?IPprimeD6:=Subgroup(F,[F.2^2]);;9gap>?K:=PreImage(projection,IPprimeD6);;9gap>?Gamma3K:=CommutatorSubgroup(K,CommutatorSubgroup(K,K));;9gap>?quotient:=G/Gamma3K;;9gap>?S:=SylowSubgroup(quotient,3);;9gap>?N:=NormalClosure(quotient,S);;9gap>?localization:=quotient/N;9Pcp?groupwithorders[2,2,2]9gap>?PrintPcpPresentation(localization);9pcp?presentationongenerators[g1,g2,g3]9g1?^2=identity9g2?^2=g39g3?^2=identity9g2?^g1=g2*g39g2?^g1^-1=g2*g3In thiscase,weseethatLGTP=DD4 .The readercan checkthattheseresultscoincidewiththoseobtainedin[DM01].Notealsothatweusedasome whatdi erentschemetocomputethislocalizationthantheoneusedin[DM01].W37einvitethereadertocheckthesamecomputations, tracinge٠xactlythestepsmadein[DM01]. W p C NBibliographGyV
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