How do I create a permutation group from the Small Group Library?

Question

If I wanted to take SmallGroup(63,3) and turn this into a permutation group, how would I do this?

I am trying to confirm some results from a research paper, but am struggling at the first hurdle here.

Answer 1

Let the group act on itself by right multiplication, as in Cayley's theorem. This gives an embedding into $S_{|G|}$.

Answer 2

The command IsomorphismPermGroup( G ) will construct an isomorphism from a given group $G$ to a subgroup of $S_{|G|}$. You can then work with the image of this isomorphism. Example code:

gap> G := SmallGroup( 63, 3 );;
gap> iso := IsomorphismPermGroup( G );;
gap> H := Image( iso );
<permutation group of size 63 with 3 generators>
gap> GeneratorsOfGroup( H );
[ (1,2,5)(3,6,11)(4,15,39)(7,21,27)(8,13,19)(9,23,47)(10,33,12)(14,29,35)(16,31,54)(17,41,20)(18,51,48)(22,37,43)(24,
    56,36)(25,49,28)(26,58,55)(30,45,42)(32,61,44)(34,62,60)(38,53,50)(40,63,52)(46,59,57),
  (1,3,8)(2,6,13)(4,9,16)(5,11,19)(7,14,22)(10,17,25)(12,20,28)(15,23,31)(18,26,34)(21,29,37)(24,32,40)(27,35,43)(30,
    38,46)(33,41,49)(36,44,52)(39,47,54)(42,50,57)(45,53,59)(48,55,60)(51,58,62)(56,61,63),
  (1,4,10,18,27,36,45)(2,7,15,24,33,42,51)(3,9,17,26,35,44,53)(5,12,21,30,39,48,56)(6,14,23,32,41,50,58)(8,16,25,34,
    43,52,59)(11,20,29,38,47,55,61)(13,22,31,40,49,57,62)(19,28,37,46,54,60,63) ]

You may also want to look at the SmallerDegreePermutationRepresentation command. It may be able to procure an isomorphism to a subgroup of $S_m$ with $m < |G|$, e.g.

gap> iso2 := SmallerDegreePermutationRepresentation( H );;
gap> Image( iso2 );
Group([ (5,6,8)(7,10,9), (1,2,3), (4,5,6,7,8,9,10) ])