We prove that a quotient G/N of a subgroup G of Sym(n) by a nonabelian minimal normal subgroup N of G embeds into Sym(m) for some $m<n$. Furthermore, if G is transitive then we can take m \le 2n/5.
Derangements in wreath products of permutation groups
The first-order theory of $\ell$-permutation groups
Growth in linear algebraic groups and permutation groups: towards a unified perspective
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