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

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.

• The naive way is just to find all subgroups of $G$ and then to find all of their indexes in $G$. – Derek Holt Mar 27 '20 at 8:07
• Just to add to Derek Holt’s clear recommendation: you will only need subgroups up to conjugacy. Tho have the action faithful, you need that the subgroup does not contain a nontriviial normal subgroup (respectively that its core is trivial). – ahulpke Mar 27 '20 at 13:11
• Thank you for your comments. But I'm looking for a direct command or a program in GAP which optimizes the time for obtaining the element structure as the product of disjoint cycles in every representation of the group $G$. This command must first find core-free subgroups, then their cosets in $G$, then show the structure of elements of the obtained representation as the product of disjoint cycles. – M. R. Mar 27 '20 at 16:35
• For example, what can we say for the element structures of the group Suzuki(32) in its different permutation representations? PSL(2,9)?... – M. R. Mar 27 '20 at 16:39

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.
• AllSubgroups uses the subgroup lattice. But it would be sufficient to only use conjugacy representatives: H:=List(ConjugacyClassesSubgroups(G),Representative); – ahulpke Mar 30 '20 at 20:35