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Is there a nice description of all divisors of $n!$ which can be realized as cardinals of permutation groups acting transitively on ${1,\dots,n}$?

A necessary condition is of course that such a divisor is a multiple of $n$.

$n$ (cyclic) and $(2n)$ dihedral 2n$(dihedral) are always possible but, without mistake on my behalf,$3n$is not feasible unless$n\equiv 1\pmod 3$(it can then be realized as a semi-direct product analogous to the dihedral group). can only be realized if$3$divides the number of invertible integers modulo$n$. 2 added 96 characters in body; deleted 5 characters in body Is there a nice description of all divisors of$n!$which can be realized as cardinals of permutation groups acting transitively on${1,\dots,n}$? A necessary condition is of course that such a divisor is a multiple of$n$.$n$(cyclic) and$(2n)$dihedral are always possible but, without mistake on my behalf,$3n$is not feasible unless$n\equiv 1\pmod 3$(it can then be realized as a semi-direct product analogous to the dihedral group). can only be realized if$3$divides the number of invertible integers modulo$n\$.

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