Mathieu groups

Remarks

The Mathieu groups are highly multiply transitive. $M_{11}$ is sharply $4$-transitive on $11$ points, and $M_{12}$ is sharply $5$-transitive on $12$ points. $M_{22}$ is $3$-transitive on $22$ points, $M_{23}$ is $4$-transitive on $23$ points, and $M_{24}$ is $5$-transitive on $24$ points. Apart from the symmetric and alternating groups, $M_{11}$, $M_{12}$, $M_{23}$ and $M_{24}$ are the only finite $4$-transitive groups. The stabiliser 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 $M_{11}$. $M_{24}$ is the automorphism group 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 $S(5, 6, 12)$. They were the first sporadic simple groups to be found, by Émile Mathieu in the nineteenth century.