The functions DCEPerm and DCEPerms have already been described, while
elementary information (such as the numbers of single and double cosets)
can be read directly from the DCE Universe produced by an
enumeration. When the number of single cosets is large, however, as in
the example of HN: 2 above, DCEPerm requires an improbably large
amount of space, so permutations cannot sensibly be obtained. However
some analysis of the permutation representation is possible directly from
the double coset table.
Specifically, functions exist to study the orbits of H, and compute their sizes and the collapsed adjacency matrices of the orbital graphs. The performance of these functions depends crucially on the size of the group M = H cap K, which will always be the muddle group of the first double coset HK. When M=K, so that K le H, then each orbit of H is just a union of double cosets and the algorithms are fast, whereas when M=1 there no benefit over extracting permutations.
GAP 3.4.4