[GAP Forum] Finding or determining existence equivalence permutation representations?

[Frederic Vanhove]

Dear forum,

thanks with the help yesterday on permutation representations.

I had another related question: suppose I have some groups (let us 
assume of the same size  but not necessarily isomorphic), represented by 
permutations on sets of the same size.
Is there any way to determine if the representations are equivalent and 
if so, to actually see the equivalence?

To be specific, in this example:

q:=3;

h:=DirectProduct(CyclicGroup(q),CyclicGroup(q));
gtemp:=AutomorphismGroup(h);
Difference(Elements(h),[Elements(h)[1]]);
g:=Action(gtemp,Difference(Elements(h),[Elements(h)[1]]),OnPoints);

w:=AllTransitiveGroups(DegreeOperation,q^2-1,Size,(q^2-1)*q*(q-1));

How could I see which group action in w is equivalent to g?
Would this also be possible for much larger groups?

Thanks,
Kind regards,
Frédéric