[GAP Forum] extending groups to isomorphics acting on a multiple of domain

[Thomas Breuer]

Dear GAP Forum,

Rudolf Zlabinger wrote:

> I try to extend the number of points acted on by permutation groups to
> multiples of the original domain.
> For example i tried to find a subgroup of S60 isomorphic to A5, thus
> acting on 60 points.
> As the samples attached show, I did it by brute force, extending the
> generators by acting on all residue classes mod the multiplier.
> 
> This method yields subgroups of the desired target symmetric groups, but
> only in one subgroup conjugacy class. The conjugacy class(es) I am
> looking for, are those, whose subgroups are transitive, as needed, for
> example, to be true rotation groups acting on the vertices of a solid.
> 
> This was possible in the case of A5 extending to S12, but not to S30 or
> even S60, as those targets are to big for executing, for example
> IsomorphicSubgroups.
> 
> In the case of finite rotation groups there are other methods using
> presentations in Linear Algebra too. I used blueprints of solids to get
> the right generator permutations for S12 and S60 for the desired rotation
> groups. 
> 
> 1. Is there a simpler method for extending the domain acted on to a
> multiple in general?
> 
> 2. Is there a feasable method for big Symmetric Groups as S60 to find
> conjugacy classes of isomorphic subgroups (for example for A5) beeing
> transitive?

If I understand the request correctly then the aim is to construct
transitive permutation representations of a given (permutation) group,
of prescribed degree.
These representations are parametrized by conjugacy classes of subgroups
of the corresponding index:
The permutation domain can be identified with the set of right cosets
of any subgroup in the class.

So all transitive permutation representations of the alternating group
$A_5$ of degrees $12$, $30$, and $60$ can be obtained by taking
representatives of classes of subgroups of the orders $5$, $2$, and $1$,
and then considering the action on the right cosets.
(In GAP, one can actually act on the return value of `RightTransversal'.)
In particular, there is only one transitive action on $60$ points,
up to renumbering of the points.

All the best,
Thomas Breuer