[GAP Forum] Permutationgroups

[Stefan Kohl]

On Sun, July 27, 2014 5:17 pm, Kurt Ewald wrote:

> how can I  construct a Permutationgroup with given Order and Degree.
>
> for instance: GroupOrder 12 and Degree 5, I found by accident
>
> G:=<(1,2,3),(1,5,2)>

You can find all transitive permutation groups of order n and degree d <= 30 by

    AllTransitiveGroups(DegreeAction,d,Size,n);

For example:

gap> AllTransitiveGroups(DegreeAction,6,Size,60);
[ L(6) = PSL(2,5) = A_5(6) ]
gap> AllTransitiveGroups(DegreeAction,6,Size,72);
[ F_36(6):2 = [S(3)^2]2 = S(3) wr 2 ]
gap> AllTransitiveGroups(DegreeAction,8,Size,32);
[ [1/4.cD(4)^2]2, 1/2[2^4]4, [4^2]2, E(8):E_4=[2^2]D(4), E(8):4=[1/4.eD(4)^2]2,
  [2^3]4, 1/2[2^4]E(4)=[1/4.dD(4)^2]2, E(8):D_4=[2^3]2^2 ]
gap> AllTransitiveGroups(DegreeAction,24,Size,240);
[ t24n570, t24n571, t24n572, t24n573, t24n574, t24n575, t24n576, t24n577, t24n578 ]

The data is taken from the GAP Transitive Groups Library,
cf. http://www.gap-system.org/Datalib/trans.html.

If -- as your example suggests -- you are also interested in
intransitive groups, you maybe need to do some programming yourself --
though for very small degrees you can simply use the following
brute force approach:

PermGroupsOfGivenDegreeAndOrder := function ( d, n )

  return Filtered(List(ConjugacyClassesSubgroups(SymmetricGroup(d)),
                       Representative),
                  G->Size(G)=n);
end;

For example:

gap> PermGroupsOfGivenDegreeAndOrder(6,12);
[ Group([ (3,6,4), (3,4)(5,6) ]), Group([ (1,5,3)(2,6,4), (1,2)(3,4) ]),
  Group([ (1,3)(4,6), (1,4)(2,6)(3,5) ]), Group([ (5,6), (2,3)(4,6) ]) ]
gap> List(last,Size);
[ 12, 12, 12, 12 ]

By the way -- your example does not have degree 5, since 4 is fixed:

gap> DegreeAction(Group((1,2,3),(1,5,2)));
4

Hope this helps,

    Stefan Kohl

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