Dear GAP-forum,

I am looking for the fastest method to decide if two
permutation groups are conjugated in a sufficiently
large symmetric group and if so to find a conjugating
permutation. Does someone know a method which does
not rely on character theory or brute force?
In other words: Given permutation groups G, H, solve
the problem

S := SymmetricGroup(
       Maximum( PermGroupOps.LargestMovedPoint(G),
                PermGroupOps.LargestMovedPoint(H) )
     );
x := RepresentativeOperation(
       S,
       AsSubgroup(S, G),
       AsSubgroup(S, H)
     );

(without computing the orbit of G, of course).

Sebastian Egner.