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  "body": "\n## Description\n\nThe Mathieu groups are five finite [[simple groups|0YMYMNQ0QZ]], written $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$.\nEach $M_n$ is a permutation group on $n$ points.\nTheir orders are:\n\n1. $\\abs{M_{11}} = 7920 = 2^4 \\cdot 3^2 \\cdot 5 \\cdot 11$,\n2. $\\abs{M_{12}} = 95040 = 2^6 \\cdot 3^3 \\cdot 5 \\cdot 11$,\n3. $\\abs{M_{22}} = 443520 = 2^7 \\cdot 3^2 \\cdot 5 \\cdot 7 \\cdot 11$,\n4. $\\abs{M_{23}} = 10200960 = 2^7 \\cdot 3^2 \\cdot 5 \\cdot 7 \\cdot 11 \\cdot 23$,\n5. $\\abs{M_{24}} = 244823040 = 2^{10} \\cdot 3^3 \\cdot 5 \\cdot 7 \\cdot 11 \\cdot 23$.\n\nThey are sporadic groups: they belong to none of the infinite families of finite simple groups.\n\n## Remarks\n\nThe Mathieu groups are highly [[multiply transitive|BDJ1AX4XBB]].\n$M_{11}$ is sharply $4$-transitive on $11$ points, and $M_{12}$ is sharply $5$-transitive on $12$ points.\n$M_{22}$ is $3$-transitive on $22$ points, $M_{23}$ is $4$-transitive on $23$ points, and $M_{24}$ is $5$-transitive on $24$ points.\nApart from the symmetric and alternating groups, $M_{11}$, $M_{12}$, $M_{23}$ and $M_{24}$ are the only finite $4$-transitive groups.\nThe [[stabiliser|0AYMP41THN]] of a point in $M_{24}$ is $M_{23}$, and the stabiliser of a point in $M_{23}$ is $M_{22}$; likewise the stabiliser of a point in $M_{12}$ is $M_{11}$.\n$M_{24}$ is the [[automorphism group|0K1SCPBNKJ]] of the Steiner system $S(5, 8, 24)$ and of the extended binary Golay code, and $M_{12}$ is the automorphism group of the Steiner system $S(5, 6, 12)$.\nThey were the first sporadic simple groups to be found, by Émile Mathieu in the nineteenth century.\n\n## Explanations\n\n### L1 human\n\nMost finite simple groups come in infinite families, like the alternating groups.\nThe Mathieu groups are five exceptions that do not fit any family.\nThey are permutation groups on $11$, $12$, $22$, $23$ and $24$ points, and they are surprisingly symmetric.\nFor example, $M_{24}$ can move any $5$ of its $24$ points to any other $5$ points, in any order.\nThey are also the symmetry groups of remarkable combinatorial objects, such as the Golay code used in error correction.\n",
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