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.

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    $\begingroup$ The naive way is just to find all subgroups of $G$ and then to find all of their indexes in $G$. $\endgroup$ – Derek Holt Mar 27 '20 at 8:07
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    $\begingroup$ 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). $\endgroup$ – ahulpke Mar 27 '20 at 13:11
  • $\begingroup$ 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. $\endgroup$ – M. R. Mar 27 '20 at 16:35
  • $\begingroup$ For example, what can we say for the element structures of the group Suzuki(32) in its different permutation representations? PSL(2,9)?... $\endgroup$ – M. R. Mar 27 '20 at 16:39

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.

  • $\begingroup$ Thank you for your comment. $\endgroup$ – M. R. Mar 30 '20 at 16:33
  • $\begingroup$ According to your comment, one can obtain all permutation representations of a specific group. For example: gap> G:=AlternatingGroup(4);; gap> H:=AllSubgroups(G);; gap> b:=List(Filtered(H,H->Size(Core(G,H))=1));; gap> s:=FactorCosetAction(G,b[1]);; gap> List(Elements(Image(s)),CycleStructurePerm); But I didn't use the Subgroup lattice and I think I calculated Isomorphism Permutation representations, which is extra working. Could you please revise my program? $\endgroup$ – M. R. Mar 30 '20 at 16:46
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    $\begingroup$ AllSubgroups uses the subgroup lattice. But it would be sufficient to only use conjugacy representatives: H:=List(ConjugacyClassesSubgroups(G),Representative); $\endgroup$ – ahulpke Mar 30 '20 at 20:35
  • $\begingroup$ Thank you very much for your answer. $\endgroup$ – M. R. Mar 30 '20 at 22:12

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