Find all possible degrees of a group in its faithful and transitive permutation representation

Question

Suppose that $G$ is a finite group of order n. We know that the action of $G$ on the cosets of its core-free subgroups is a faithful and transitive action and the degree in this permutation representation of $G$ is $(G:H)$. I need a GAP program or command to find all possible degrees of a group $G$ in its faithful and transitive permutation representation.

In fact, I need a command or program in GAP to find the cycle structure of elements of a group $G$ in all of its permutation representations.

Answer 1

There is (not surprising) no ready made functions for this, but this is not hard to code on your own. What I suggest is to:

1. Find all faithful transitive permutation representations from the lattice and
2. Use these representations to find the cycle structures. (I don't think you will gain much by avoiding the representations for the groups you mention).

For 1., calculate the subgroup lattice, take representatives of the classes. Identify wich subgroups $U$ satisfy that Size(Core(G,U))=1. Then for each such subgroup use FactorCosetAction to get the permutation action on the cosets.