[GAP Forum] replacing group action by permutations of [1..n]

[Stefan Kohl]

On Wed, November 28, 2012 2:30 pm, Stefan Kohl wrote:
> Dear Frederic,
>
>> in the following two examples, GAP constructs a group of permutations,
>> not represented as elements of the symmetric group S_n:
>>
>> h:=PSL(2,5);
>> g:=AutomorphismGroup(h);
>> Elements(g);
>>
>> gf:=GF(64);
>> g:=GaloisGroup(gf);
>> Elements(g);
>>
>> However, I would like to have an equivalent representation on
>> respectively [1..60] and [1..64].
>>
>> Is this possible?
>
> Yes. -- You can do the following:
>
> gap> h:=PSL(2,5);
> Group([ (3,5)(4,6), (1,2,5)(3,4,6) ])
> gap> g:=AutomorphismGroup(h);
> <group of size 120 with 3 generators>
> gap> g_regular := Action(g,AsList(g),OnRight);
> <permutation group with 3 generators>
> gap> DegreeAction(g_regular);
> 120

Ah, I see you asked for the action on 60 points -- so do:

gap> gp := Action(g,h,OnPoints);
<permutation group with 3 generators>
gap> DegreeAction(gp); # 60 - 1, identity is fixed by every automorphism
59

    Stefan