[GAP Forum] easy way to generate permutation groups?

[R. Keith Dennis]

Hi, I'm interested in studying certain groups which arise as
permutations of subsets of a given group.  As a simple example, let G
be any finite group, and S = G x G, the set of pairs of elements of G.
Define a to be the permutation of S generated by (x,y) --> (x,xy) and
b to be the one generated by (x,y) --> (yx,y).  Is there a simple way
to construct the subgroup of the permutation group on S generated by a
and b?  In this case both a and b have order exp(G) & for example I'd
like to be able to compute a presentation for the group of
permutations they generate.  More generally I'd like to study other
operations on certain subsets (or sequences) of elements derived from
a fixed group G.

As permutation groups seem to be given as permutations of sets of
integers, it almost seems that I should (in essence) have to describe
a one-to-one correspondence of S with a set of integers [1..m] and 
describe a and b by explicitly computing via this correspondence.
Is that sort of thing really necessary?  It usually seems that there
are built-in operations in GAP to avoid such.

Perhaps I'm missing something obvious.  Suggestions most welcome!

Keith