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This is a follow-up to normal form for some finite groups, extending the small groups library.

Not being familiar with groups, I wonder whether it is possible to check efficiently whether a group (given as a permutation group) is isomorphic to a generalized symmetric group.

Initial computer experiments indicate that the parameter $m$ in $\mathbb Z_m\wr\mathfrak S_n$ might be twice the index of the derived subgroup in the group.

From a practical point of view, I am trying to do this with GAP.

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    $\begingroup$ How is the "given group" given? As just some arbitrary permutation group? As a matrix group? By generators and relations? Those are three very different situations from an algorithmic standpoint. The third is in fact an uncomputable problem. $\endgroup$
    – Johannes Hahn
    Commented Dec 15, 2017 at 21:10
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    $\begingroup$ You could use $\mathtt{RadicalGroup}$ to find the largest solvable normal subgroup (or $\mathtt{FittingSubgroup}$ to find the largest nilpotent normal subgroup) and then try to identify this as $C_m^n$, and the quotient as $S_n$. You might find it more productive to use the GAP Forum mailing list to ask how to do specific things in GAP. $\endgroup$
    – Derek Holt
    Commented Dec 15, 2017 at 21:55
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    $\begingroup$ Yes I was probably assuming that $n \ge 5$. I would expect that for $n \le 4$ you could do it by testing directly for isomorphism, at least for reasonably small $m$. $\endgroup$
    – Derek Holt
    Commented Dec 16, 2017 at 8:51
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    $\begingroup$ I am unwilling to make this into an answer, because I don't really believe that this question is suitable for a forum devoted to research level mathematics. I appreciate that you are not a specialist in group theory, but what you seem to be asking for is advice on how to perform certain calculations in GAP, and there are better places available for that. Also, it would be helpful to have some idea of the range of $m$ and $n$ that you have in mind. $\endgroup$
    – Derek Holt
    Commented Dec 16, 2017 at 18:00
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    $\begingroup$ It's OK for me if the question is closed, although I must say that I was actually not asking for advice on how to use GAP. I would have been happy with a mathematical answer. In particular, I still do not know how to recognise $C_m \wr S_3$ and $C_m \wr S_4$ other than testing for isomorphism. $\endgroup$
    – Martin Rubey
    Commented Dec 21, 2017 at 20:08

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