Quasi canonical isomorphisms between finite groups and subgroups of the symmetric group?

Question

I'm implementing finite groups in Forth for my blog and since any group is a subgroup of a permutation group it make sense to let the standard elements be permutation schemas:

123  12345
231  43215

abbreviated $231$, $43215$ and so on. But when dealing with quote groups the elements become (co-) sets of permutations. It would be very nice to be able to transform those quote groups to isomorphic permutation groups.

Are there methods to find an isomorphic permutation group for any finite group, or at least find the smallest symmetric group that contains an isomorphic subgroup? Say from an arbitrary group of co-sets of permutations.

Answer 1

To find an isomorphic permutation group simply take the action on the elements (Cayley's theorem). In most cases this is far from optimal.

If you want a representation of minimal degree you probably cannot do better in the worst-case than a brute-force approach by running through sets of subgroups. This looks infeasible.

The system GAP for example has a function SmallerDegreePermutationRepresentation that uses many heuristics to find a middle way, but this can still fail in many cases.

There is theoretical work that considers the minimal faithful permutation degree for certain groups, but it covers only rather specific subclasses of groups.