How do I construct … in GAP?

This page is devoted to answering some basic questions along the line
“How do I construct … in GAP?” You may view the html source code
for the GAP commands without the output or GAP prompt.

Please send suggestions, additions, corrections to
David Joyner.

This page itself is under construction…

Questions

How
do I construct a … group?
permutation
dihedral 
cyclicconjugacy classes of a
finitely presented
How
do I … a polynomial?
How do I find the … of a group
representation?
How
do I compute an mod m, where A is …?
Given
a group G, how do I compute … ?

Answers

    • permutation:
      To construct a permutation group, write down generators in disjoint cycle notation,
      put them in a list (i.e., surround them by square brackets), andThe permutation group G generated by the cycles
      (1,2)(3,4) and (1,2,3):
gap> G:=Group((1,2)(3,4),(1,2,3));

Group([ (1,2)(3,4), (1,2,3) ])

This is of course a subgroup of the symmetric group S4 on 4
letters.
Indeed, this G is in fact the alternating group
on four letters, A4.

By virtue of the fact that the permutations generating G employ
integers less than or equal to 4, this group G
is a subgroup of the symmetric group S4 on 4
letters. Some permutation groups have special constructions:

gap> S4:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> A4:=AlternatingGroup(4);
Alt( [ 1 .. 4 ] )
gap> IsSubgroup(S4,G);
true
gap> IsSubgroup(A4,G);
true
gap> S3:=SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> IsSubgroup(S3,G);
false

    • dihedral
      To construct a dihedral group, use the special “DihedralGroup” command:
gap> G:=DihedralGroup(6);

gap> Size(G);
6
gap> f:=GeneratorsOfGroup( G );
[ f1, f2 ]
gap> f[1]^2; f[2]^3;
identity of ...
identity of ...
gap> f[1]^2= f[2]^3;
true
  • cyclic group
    To construct a cyclic group, you may
    construct integers mod n:

    gap> R:=ZmodnZ( 12);
(Integers mod 12)
gap> a:=Random(R);
ZmodnZObj( 11, 12 )
gap> 4*a;
ZmodnZObj( 8, 12 )
gap> b:=Random(R);
ZmodnZObj( 9, 12 )
gap> a+b;
ZmodnZObj( 8, 12 )

    or use the special “CyclicGroup” command

    gap> G:=CyclicGroup(12);
pc group of size 12 with 3 generators
gap> a:=Random(G);
f3^2
gap> f:=GeneratorsOfGroup( G );
[ f1, f2, f3 ]
gap> f[1]^4;
f3
gap> f[1]^12;
identity of ...

  • conjugacy:
    The conjugacy classes of a group G are computed using
    the “ConjugacyClasses” command. This is a list
    of classes{x^-1*g*x | x in G}.

    gap> G:=SL(2,7);
SL(2,7)
gap> CG:=ConjugacyClasses(G);
[ [ [ Z(7)^0, 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ]^G,
  [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^5 ] ]^G,
  [ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^5 ] ]^G,
  [ [ Z(7)^3, 0*Z(7) ], [ 0*Z(7), Z(7)^3 ] ]^G,
  [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^2 ] ]^G,
  [ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^2 ] ]^G,
  [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ]^G,
  [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^4 ] ]^G,
  [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7) ] ]^G,
  [ [ Z(7)^4, 0*Z(7) ], [ 0*Z(7), Z(7)^2 ] ]^G,
  [ [ Z(7)^5, 0*Z(7) ], [ 0*Z(7), Z(7) ] ]^G ]
gap> g:=Representative(CG[3]); Order(g);
[ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^5 ] ]
14
gap> g:=Representative(CG[4]); Order(g);
[ [ Z(7)^3, 0*Z(7) ], [ 0*Z(7), Z(7)^3 ] ]
2
gap> g:=Representative(CG[5]); Order(g);
[ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^2 ] ]
7
gap> g:=Representative(CG[6]); Order(g);
[ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^2 ] ]
7
gap>

  • presented
    To construct a finitely presented group in GAP, use the
    “FreeGroup” and “” commands. Here is one example.

    gap> M12 := MathieuGroup( 12 );
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ])
gap> F := FreeGroup( "a", "b", "c" );
free group on the generators [ a, b, c ]
gap> words := [ F.1, F.2 ];
[ a, b ]
gap> P := PresentationViaCosetTable( M12, F, words );
presentation with 3 gens and 10 rels of total length 97
gap> TzPrintRelators( P );
#I  1. c^2
#I  2. b^4
#I  3. a*c*a*c*a*c
#I  4. a*b^2*a*b^-2*a*b^-2
#I  5. a^11
#I  6. a^2*b*a^-2*b^2*a*b^-1*a^2*b^-1
#I  7. a*b*a^-1*b*a^-1*b^-1*a*b*a^-1*b*a^-1*b^-1
#I  8. a^2*b*a^2*b^2*a^-1*b*a^-1*b^-1*a^-1*b^-1
#I  9. a*b*a*b*a^2*b^-1*a^-1*b^-1*a*c*b*c
#I  10. a^4*b*a^2*b*a^-2*c*a*b*a^-1*c
gap> G := FpGroupPresentation( P );
fp group on the generators [ a, b, c ]
gap> RelatorsOfFpGroup( G );  
[ c^2, b^4, a*c*a*c*a*c, a*b^-2*a*b^-2*a*b^-2, a^11, a^2*b*a^-2*b^-2*a*b^-1*a^2*b^-1, a*b*a^-1*b*a^-1*b^-1*a*b*a^-1*b*a^-1*b^-1,
  a^2*b*a^2*b^-2*a^-1*b*a^-1*b^-1*a^-1*b^-1, a*b*a*b*a^2*b^-1*a^-1*b^-1*a*c*b*c, a^4*b*a^2*b*a^-2*c*a*b*a^-1*c ]
gap> Size(M12);
95040
gap> Size(G);
95040
gap> IsomorphismGroups(G,M12); 
????????

    The last command is computationally intensive and requires more
    than the default memory allocation of 256M of RAM.

    Here is another example.

    gap> F := FreeGroup( "a", "b");
free group on the generators [ a, b ]
gap> G:=F/[F.1^2,F.2^3,F.1*F.2*F.1^(-1)*F.2^(-1)];
fp group on the generators [ a, b ]
gap> Size(G);
6

  • rref
    The key command for row reduction is “TriangulizeMat”.
    The following example illustrates the syntax.

    gap> M:=[[1,2,3,4,5],[1,2,1,2,1],[1,1,0,0,0]];
[ [ 1, 2, 3, 4, 5 ], [ 1, 2, 1, 2, 1 ], [ 1, 1, 0, 0, 0 ] ]
gap> TriangulizeMat(M);
gap> M;
[ [ 1, 0, 0, -1, 1 ], [ 0, 1, 0, 1, -1 ], [ 0, 0, 1, 1, 2 ] ]
gap> Display(M);
[ [   1,   0,   0,  -1,   1 ],
  [   0,   1,   0,   1,  -1 ],
  [   0,   0,   1,   1,   2 ] ]
gap> M:=Z(3)^0*[[1,2,3,4,5],[1,2,1,2,1],[1,1,0,0,0]];
[ [ Z(3)^0, Z(3), 0*Z(3), Z(3)^0, Z(3) ],
  [ Z(3)^0, Z(3), Z(3)^0, Z(3), Z(3)^0 ],
  [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ] ]
gap> TriangulizeMat(M);
gap> Display(M);
 1 . . 2 1
 . 1 . 1 2
 . . 1 1 2
gap>

  • kernel:
    There are different methods for matrices over the integers and
    matrices over a field.For integer entries, related commands include
    “NullspaceIntMat” and “SolutionNullspaceIntMat”
    in section

    25.1 “Linear equations over the integers and Integral Matrices”

    of the reference manual.

    gap> M:=[[1,2,3],[4,5,6],[7,8,9]];
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
gap> NullspaceIntMat(M);
[ [ 1, -2, 1 ] ]
gap> SolutionNullspaceIntMat(M,[0,0,1]);
[ fail, [ [ 1, -2, 1 ] ] ]
gap> SolutionNullspaceIntMat(M,[0,0,0]);
[ [ 0, 0, 0 ], [ [ 1, -2, 1 ] ] ]
gap> SolutionNullspaceIntMat(M,[1,2,3]);
[ [ 1, 0, 0 ], [ [ 1, -2, 1 ] ] ]

    Here (0,0,1) is not in the image of M
    (under v-> v*M) but (0,0,0) and (1,2,3) are.

    For field entries, related commands include
    “NullspaceMat” and “TriangulizedNullspaceMat”
    in section

    24.6 “Matrices Representing Linear Equations and the Gaussian Algorithm”

    of the reference manual.

    gap> M:=[[1,2,3],[4,5,6],[7,8,9]];
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
gap> NullspaceMat(M);
[ [ 1, -2, 1 ] ]
gap> TriangulizedNullspaceMat(M);
[ [ 1, -2, 1 ] ]
gap> M:=[[1,2,3,1,1],[4,5,6,1,1],[7,8,9,1,1],[1,2,3,1,1]];
[ [ 1, 2, 3, 1, 1 ], [ 4, 5, 6, 1, 1 ], [ 7, 8, 9, 1, 1 ], 
  [ 1, 2, 3, 1, 1 ] ]
gap> NullspaceMat(M);
[ [ 1, -2, 1, 0 ], [ -1, 0, 0, 1 ] ]
gap> TriangulizedNullspaceMat(M);
[ [ 1, 0, 0, -1 ], [ 0, 1, -1/2, -1/2 ] ]

  • characteristic polynomial:
    Please see section
    24.12.1 of the GAP reference manual
    for examples of characteristic polynomial of a
    square matrix (“CharacteristicPolynomial”) and
    section

    56.3     for examples of the “characteristic polynomial”
    (called a “TracePolynomial”) of an
    element of a field extension.

  • character:
    GAP contains very extensive character theoretic functions
    and data libraries (including an interface the character table in the
    Atlas).
    Here is just one simple example.

    gap> G:=Group((1,2)(3,4),(1,2,3));
Group([ (1,2)(3,4), (1,2,3) ])
gap> T:=CharacterTable(G);
CharacterTable( Alt( [ 1 .. 4 ] ) )
gap> Display(T);
CT1

     2  2  2  .  .
     3  1  .  1  1

       1a 2a 3a 3b
    2P 1a 1a 3b 3a
    3P 1a 2a 1a 1a

X.1     1  1  1  1
X.2     1  1  A /A
X.3     1  1 /A  A
X.4     3 -1  .  .

A = E(3)^2
  = (-1-ER(-3))/2 = -1-b3
gap> irr:=Irr(G);
[ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] ),
  Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3), E(3)^2 ] ),
  Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, -1, 0, 0 ] ) ]
gap> Display(irr);
[ [       1,       1,       1,       1 ],
  [       1,       1,  E(3)^2,    E(3) ],
  [       1,       1,    E(3),  E(3)^2 ],
  [       3,      -1,       0,       0 ] ]
gap> chi:=irr[2]; gamma:=CG[3]; g:=Representative(gamma); g^chi;
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] )
(1,2,3)^G
(1,2,3)
E(3)^2

    For further details and examples, see chapters
    69–
    72 of the
    GAP reference manual.

  • brauer:
    Just a simple example of what GAP can do here.
    To construct a Brauer character table:

    gap> G:=Group((1,2)(3,4),(1,2,3));
Group([ (1,2)(3,4), (1,2,3) ])
gap> irr:=IrreducibleRepresentations(G,GF(7));
[ [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^0 ] ] ],
  
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^4 ] ] ],
  
[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^2 ] ] ],
  
[ (1,2)(3,4), (1,2,3) ] -> [
      
[ [ 0*Z(7), Z(7)^3, Z(7)^0 ], [ 0*Z(7), Z(7)^3, 0*Z(7) ], 
[ Z(7)^0, Z(7)^3, 0*Z(7) ] ],
      [ [ 0*Z(7), Z(7)^0, 0*Z(7) ], 
[ 0*Z(7), 0*Z(7), Z(7)^0 ], [ Z(7)^0, 0*Z(7), 0*Z(7) ] ]
     
] ]
gap> brvals := List(irr,chi-> List(ConjugacyClasses(G),c->
BrauerCharacterValue(Image(chi, Representative(c)))));
[ [ 1, 1, 1, 1 ], [ 1, 1, E(3)^2, E(3) ], [ 1, 1, E(3), E(3)^2 ], 
[ 3, -1, 0, 0 ] ]
gap> Display(brvals);
[ [       1,       1,       1,       1 ],
  
[       1,       1,  E(3)^2,    E(3) ],
  
[       1,       1,    E(3),  E(3)^2 ],
  
[       3,      -1,       0,       0 ] ]
gap>

    List(ConjugacyClasses(G),c->BrauerCharacterValue(Image(chi, Representative(c)))));
    #Display(brvals);
    T:=CharacterTable(G);
    Display(T);
    –>

  • polynomial
    There are various ways to construct a polynomial in GAP.

    gap> Pts:=Z(7)^0*[1,2,3];
[ Z(7)^0, Z(7)^2, Z(7) ]
gap> Vals:=Z(7)^0*[1,2,6];
[ Z(7)^0, Z(7)^2, Z(7)^3 ]
gap> g:=InterpolatedPolynomial(GF(7),Pts,Vals);
Z(7)^5*x_1^2+Z(7)

    Or:

    gap> p:=3;; F:=GF(p);;
gap> R:=PolynomialRing(F,["x1","x2"]);
PolynomialRing(..., [ x1, x2 ])
gap> vars:=IndeterminatesOfPolynomialRing(R);;
gap> x1:=vars[1]; x2:=vars[2];
x1
x2
gap> p:=x1^5-x2^5;
x1^5-x2^5
gap> DivisorsMultivariatePolynomial(p,R);
[ x1^4+x1^3*x2+x1^2*x2^2+x1*x2^3+x2^4, x1-x2 ]

    Or:

    gap> x:=X(Rationals);
x_1
gap> f:=x+x^2+1;
x_1^2+x_1+1
gap> Value(f,[x],[1]);
3

  • factor
    To factor a polynomial in GAP, there is one command for
    univariate polynomials (“Factors”) and another command for
    multivariate polynomials (“DivisorsMultivariatePolynomial”).For a factoring a univariate polynomial,
    GAP provides only methods over finite fields
    and over subfields of cyclotomic fields.
    Please see the
    examples given in section

    64.10 “Polynomial Factorization”     for more details.For multivariate polynomials,
    a very slow algorithm has been implemented in GAP
    and an interface to a very fast algorithm in
    Singular
    has been implemented for those who have both Singular and
    the GAP Singular package
    installed. The former of these was
    illustrated above in
    “polynomial” above.
    (Again, the ground field must be a finite field
    or a subfields of cyclotomic fields.)
    For the latter, please see the example
    in the (GAP-)Singular manual
    FactorsUsingSingularNC.

  • roots
    There are some situtations where GAP does find the roots
    of a polynomial but GAP does not do this generally.
    (The roots must generate either a finite field
    or a subfield of a cyclotomic field.) However, there is a package called

    RadiRoot     which must be installed which does help to do this
    for polynomials with rational coefficients
    (radiroot itself requires other packages to be installed;
    please see the webpage for more details).The “Factors” command actually has an option which allows you to
    increase the groundfield so that a factorization actually
    returns the roots. Please see the
    examples given in section

    64.10 “Polynomial Factorization”     for more details.Here is a second appoach.

    gap> p:=3; n:=4; F:=GF(p^n); c:=Random(F); r:=2;
3
4
GF(3^4)
Z(3^4)^79
2
gap>  x:=X(F,1); f:=x^r-c*x+c-1;
x_1
x_1^2+Z(3^4)^39*x_1+Z(3^4)^36
gap>  F_f:=FieldExtension( F, f );
AsField( GF(3^4), GF(3^8) )
gap>  alpha:=RootOfDefiningPolynomial(F_f);
Z(3^4)^36
gap> Value(f,[x],[alpha]);
0*Z(3)

    Here is a third. First, enter the following program:

    RootOfPolynomial:=function(f,R)
 local F0,Ff,a;
 F0:=CoefficientsRing(R);
 Ff:=FieldExtension(F0,f);
 a:=RootOfDefiningPolynomial(Ff);
 return a;
end;

    Here’s how this can be used to find a root:

    gap> F:=Rationals;
Rationals
gap> x:=X(F,1); f:=x^2+x+1;
x_1
x_1^2+x_1+1
gap> R:=PolynomialRing( F, [ x ]);
PolynomialRing(..., [ x_1 ])
gap> a:=RootOfPolynomial(f,R);
E(3)
gap> # check:
gap> Value(f,[x],[a]);
0

    Related links:

    1. In the GAP Forum:

      Hensel lifting discussion      .
    2. In the manual,

      Galois groups      .
  • evaluate:
    The relevant command is “Value”. There are several examples already on
    this page. For others, please see the examples given in section
    64.7 Multivariate polynomials of the manual.
    For sparse uivariate polynomials, there is also the command
    “ValuePol” in section
    23.6 of the manual.

  • integer power
    This is easy and intuitive:

    gap> a:=1000; n:=100000; m:=123;
1000
100000
123
gap> a^n mod m;
1

  • matrix power:
    This too is easy and intuitive:

    gap> A:=[[1,2],[3,4]]; n:=100000; m:=123;
[ [ 1, 2 ], [ 3, 4 ] ]
100000
123
gap> A^n mod m;
[ [ 1, 41 ], [ 0, 1 ] ]
  • polynomial power
    GAP allows you to do arithmetic over the polynomial
    ring R[x], where R = Z/nZ (where n is a positive integer).
    Here’s an example.

    gap> Z4:=ZmodnZ(4);
(Integers mod 4)
gap> R:=UnivariatePolynomialRing(Z4,1);
PolynomialRing(..., [ x ])
gap> x:=IndeterminatesOfPolynomialRing(R)[1];
x
gap> I:=TwoSidedIdealByGenerators( R,[x^8-x^0]);
two-sided ideal in PolynomialRing(..., [ x ]), (1 generators)
gap> gen:=x^8-x^0;
x8-ZmodnZObj(1,4)
gap> QuotientRemainder(R,x^8,gen);
[ ZmodnZObj(1,4), ZmodnZObj(1,4) ]
gap> QuotientRemainder(R,x^15,gen);
[ x^7, x^7 ]
gap> QuotientRemainder(R,x^15+x^8,gen);
[ x^7+ZmodnZObj(1,4), x^7+ZmodnZObj(1,4) ]
gap> PowerMod( R, x+x^0, 15, gen );
ZmodnZObj(0,4)
gap> PowerMod( R, x, 15, gen );
x^7

  • Groebner basis
    GAP’s Groebner bases algorithms are relatively slow
    and are included mostly for simple examples and for
    teaching purposes. However, a GAP interface to a very
    fast algorithm in Singular
    has been implemented for those who have both Singular and
    the
    GAP Singular package
    installed. The former of these is
    illustrated in section
    64.17 Groebner bases of the GAP manual.
    For the latter, please see the example
    in the (GAP-)Singular manual
    GroebnerBasis.

  • normal subgroup:
    Here is an example:

    gap> G := AlternatingGroup( 5 );
Group( (1,2,5), (2,3,5), (3,4,5) )
gap> normal := NormalSubgroups( G );
[ Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [  ] ), 
  Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
    [ (1,2)(3,4), (1,3)(4,5), (1,4)(2,3) ] ) ]

    Related links:

    1. Please see Volkmar Felsch’s
      GAP Forum response to a related question.
    2. The

      xgap       package displays subgroup lattices graphically.
  • abelian subgroup
    One idea to compute all the abelian subgroups is to compute all the
    subgroups then “filter” out the abelian ones.
    Here is an illustration, taked from a
    GAP Forum response Volkmar Felsch.

    gap> G := AlternatingGroup( 5 );
Group( (1,2,5), (2,3,5), (3,4,5) )
gap> classes := ConjugacyClassesSubgroups( G );
[ ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), 
    (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [  ] ) ),
  ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), 
    (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
    [ (2,3)(4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), 
    (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
    [ (3,4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), 
    (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
    [ (2,3)(4,5), (2,4)(3,5) ] ) ), ConjugacyClassSubgroups( Group( 
    (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), 
    (3,4,5) ), [ (1,2,3,4,5) ] ) ), ConjugacyClassSubgroups( Group( 
    (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), 
    (3,4,5) ), [ (3,4,5), (1,2)(4,5) ] ) ), 
  ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), 
    (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
    [ (1,2,3,4,5), (2,5)(3,4) ] ) ), ConjugacyClassSubgroups( Group( 
    (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), 
    (3,4,5) ), [ (2,3)(4,5), (2,4)(3,5), (3,4,5) ] ) ), 
  ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Group( 
    (1,2,5), (2,3,5), (3,4,5) ) ) ]
gap> cl := classes[4];
ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), 
(3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
[ (2,3)(4,5), (2,4)(3,5) ] ) )
gap> length := Size( cl );
5
gap> rep := Representative( cl );
Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
[ (2,3)(4,5), (2,4)(3,5) ] )
gap> order := Size( rep );
4
gap> IsAbelian( rep );
true
gap> abel := Filtered( classes, cl -> IsAbelian( Representative( cl ) ) );
[ ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), 
    (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [  ] ) ),
  ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), 
    (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
    [ (2,3)(4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), 
    (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
    [ (3,4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), 
    (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), 
    [ (2,3)(4,5), (2,4)(3,5) ] ) ), ConjugacyClassSubgroups( Group( 
    (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), 
    (3,4,5) ), [ (1,2,3,4,5) ] ) ) ]
  • homology
    This depends on how the group is given. For example, suppose that
    G is a permutation group with generators genG and
    H is a permutation group with generators genH. To find a
    homomorphism from G to H, one may use the
    “GroupHomomorphismByImages” or “GroupHomomorphismByImagesNC”
    commands. For examples of the syntax, please see
    section
    38.1 Creating Group Homomorphisms.Here’s an illustration of how to convert a finitely presented
    group into a permutation group.

    gap> p:=7;
7
gap> G:=PSL(2,p);
Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8) ])
gap> H:=SchurCover(G);
fp group of size 336 on the generators [ f1, f2, f3 ]
gap> iso:=IsomorphismPermGroup(H);
[ f1, f2, f3 ] -> [ (1,2,4,3)(5,9,7,10)(6,11,8,12)(13,14,15,16),
  (2,5,6)(3,7,8)(11,13,14)(12,15,16), (1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,
    15)(14,16) ]
gap> H0:=Image(iso);                       # 2-cover of PSL2
Group([ (1,2,4,3)(5,9,7,10)(6,11,8,12)(13,14,15,16),
  (2,5,6)(3,7,8)(11,13,14)(12,15,16), (1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,
    15)(14,16) ])
gap> IdGroup(H0);
[ 336, 114 ]
gap> IdGroup(SL(2,7));
[ 336, 114 ]
gap>
  • semi-direct product(Contributed by Nilo de Roock):
    As you can easily verify, D8 is isomorphic to C2:C4. Or in GAP…

    N:=CyclicGroup(IsPermGroup,4);
G:=CyclicGroup(IsPermGroup,2);
AutN:=AutomorphismGroup(N);
f:=GroupHomomorphismByImages(G,AutN,GeneratorsOfGroup(G),[Elements(AutN)[2]]);
NG:=SemidirectProduct(G,f,N);

    Verify with

    StructureDescription(NG);
  • semi-direct products(Contributed by Nilo de Roock):
    The following shows how to construct all non-abelian groups
    of order 12 as semi-direct products. These products are not
    trivial yet small enough to verify by hand.

    #D12 = (C2 x C2) : C3
G1:=CyclicGroup(IsPermGroup,2);
G2:=CyclicGroup(IsPermGroup,2);
G:=DirectProduct(G1,G2);
N:=CyclicGroup(IsPermGroup,3);
AutN:=AutomorphismGroup(N);
f:=GroupHomomorphismByImages(G,AutN,[Elements(G)[1],Elements(G)[2],Elements(G)[3],Elements(G)[4]],[Elements(AutN)[1],Elements(AutN)[2],Elements(AutN)[1],Elements(AutN)[2]]);
NG:=SemidirectProduct(G,f,N);
Print(str(NG));
Print("\n");

    #T = C4 : C3
G:=CyclicGroup(IsPermGroup,4);
N:=CyclicGroup(IsPermGroup,3);
AutN:=AutomorphismGroup(N);
f:=GroupHomomorphismByImages(G,AutN,[Elements(G)[1],Elements(G)[2],Elements(G)[3],Elements(G)[4]],[Elements(AutN)[1],Elements(AutN)[2],Elements(AutN)[1],Elements(AutN)[2]]);
NG:=SemidirectProduct(G,f,N);
Print(str(NG));
Print("\n");

    #A4 = C3 : (C2 x C2)
G:=CyclicGroup(IsPermGroup,3);
N1:=CyclicGroup(IsPermGroup,2);
N2:=CyclicGroup(IsPermGroup,2);
N:=DirectProduct(G1,G2);
AutN:=AutomorphismGroup(N);
f:=GroupHomomorphismByImages(G,AutN,[Elements(G)[1],Elements(G)[2],Elements(G)[3]],[Elements(AutN)[1],Elements(AutN)[4],Elements(AutN)[5]]);
NG:=SemidirectProduct(G,f,N);
Print(str(NG));
Print("\n");
  • cohomology
    GAP will compute the Schur multiplier
    H2(G,C) using the
    “AbelianInvariantsMultiplier” command.
    Here is an example showing how to find H2(A5,C),
    where A5 is the alternating group on 5 letters.

    gap> A5:=AlternatingGroup(5);
Alt( [ 1 .. 5 ] )
gap> AbelianInvariantsMultiplier(A5);
[ 2 ]

    So, H2(A5,C) is Z/2Z.

    Related links:

    1. See section

      37.23       and
      section

      37.24       of the GAP manual.
    2. See D. Holt’s GAP package
      cohomolo.