[GAP Forum] building group action without losing original names?

[Frederic Vanhove]

Hello,

I have yet another question on (equivalent) group actions.
When I build an action on a set, is there way to have GAP store a 
bijection between the original points on which he acts, and the labels 
[1..v]?

For instance, when I do:

G:=SymmetricGroup(4);
g:=AutomorphismGroup(G);
act:=Action(g,G,OnPoints);
Orbits(act);

the output in the end is:
[ [ 2, 7, 5, 21, 16, 13 ], [ 3, 11, 6, 9, 20, 4, 17, 14 ],
       [ 8, 24, 23, 15, 10, 18 ], [ 12, 22, 19 ] ]

while elements of G itself look like (1,4)(2,3), and I would want the 
orbits in that form.
(This is just an example, I had other more complicated sets in mind)

Kind regards,
Frédéric