Action of a coset space in GAP

Question

I would like to get the orbit of a point $g$ under the action of a coset space ($H\backslash G$, where $H\not\unlhd G$). Specifically, I have two permutations $g,h \in S_{n}$ and I would like to find the orbit of $g$ under conjugation by elements of $C_{S_{n}}(h) \backslash S_{n}$. That is, if $k_{0} \in C_{S_{n}}(h), k_{1} \in S_{n}$, I want to consider $k_{1}gk_{1}^{-1}$ and $k_{0}k_{1}g(k_{0}k_{1})^{-1}$ as equivalent, so I don't want them to both appear in the orbit of $g$. Is it possible to use GAP's action machinery to accomplish this, or would I have to implement it by hand?

EDIT: One way to get the end goal (pairs of permutations of given cycle types, up to simultaneous conjugation by $S_{n}$), is as the double coset representatives for $C(h)\backslash S_{n}/C(g)$, but when I try to do the double cosets, I either hit the recursion limit (this happened in $S_{11}$; I doubled the limit from 5000 to 10000 and still hit it, so I didn't figure I had much hope for degrees higher than 11 using this method) or I get a "Cannot extend workspace". I was able to get what I wanted by running through the conjugacy class of $g$, but I got too many permutations (in one case where I wanted 10, I got 17,000). So my next attempt was to reduce the elements that I am iterating over, by modding out by $C(h)$.

Answer 1

Double cosets (which have some attempts built in to be clever) would be the way to go.

The double coset calculation should go through without problem -- if there is a recursion trap, it must indicate a bug In this case email me so we can find out what is happening.

I just tried a random example (in the development version of GAP which is what I have installed), which finished in a few seconds:

gap> g:=SymmetricGroup(11);
Sym( [ 1 .. 11 ] )
gap> a:=Random(g);
(1,10,7,4,2,9,5,8,3,6)
gap> b:=Random(g);
(1,5,8,10,2,9,11)(3,6)
gap> dc:=DoubleCosets(g,Centralizer(g,a),Centralizer(g,b));;
gap> Length(dc);
142560

Note: The error observed by the poster is due to an older version of GAP, it does not arise in the current release 4.8.7