It is well known that every group is isomorphic with a subgroup of some symmetric group. So I am interested with the case of the sporadic groups and I am asking if for each of them the minimal order of a symmetric group having a subgroup isomorphic with this sporadic group is known.
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7$\begingroup$ I think this is equivalent to asking for the minimal degree of a faithful permutation representation of each sporadic simple group. Such a representation is transitive and primitive.I presume these are by now tabulated, but I do not know a reference. $\endgroup$– Geoff RobinsonCommented Dec 28, 2020 at 20:33
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9$\begingroup$ You can find all of this information in the ATLAS of Finite Groups. In fact all of the maximal subgroups of all of the sporadic groups are known except for a very small number of uncertainties about whether certain groups arise as maximal subgroups of the Monster. But they all involve possible subgroups of very large index. $\endgroup$– Derek HoltCommented Dec 28, 2020 at 22:28
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4$\begingroup$ Just to spell out the details implicit in the previous comments: An embedding $G \subset S_n$ is the same as a degree-$n$ permutation representation (i.e. an action of $G$ on a set of order $n$), and minimality implies it is transitive and so this is the action of $G$ on $G/H$ for some subgroup $H \subset G$ (namely, the stabilizer of a point), and so you are asking for the subgroup of $G$ of maximal order (hence minimal $n = |G/H|$). The maximal-order subgroups of sporadic groups are known, and available for example in the ATLAS. $\endgroup$– Theo Johnson-FreydCommented Dec 28, 2020 at 22:47
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3$\begingroup$ For example, if $M_n$ is a sporadic Mathieu group (i.e. $n=11,12,22,23,24$), then $n$ is the minimal number $m$ such that $M_n \subset S_m$, because $M_{n-1}$ is a maximal subgroup of $M_n$ of maximal order, and $|M_n : M_{n-1}|=n$. Note that $M_{n-1}$ is not sporadic when $n=11,22$. $\endgroup$– Sebastien PalcouxCommented Dec 29, 2020 at 1:06
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$\begingroup$ Thank you very much for thesevinformations and explanations $\endgroup$– Gérard LangCommented Dec 29, 2020 at 16:49
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