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Related concepts
exceptional structures, exceptional isomorphisms
exceptional finite rotation groups:
and Kac-Moody groups:
exceptional Jordan superalgebra, $K_10$
The Mathieu groups, denoted $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$ are five of the sporadic finite simple groups in the Happy Family. They were first described in the 1860-70s by Émile Mathieu, and the first such groups to be discovered.
They arise as the automorphism groups of Steiner systems. The orders of the groups are as follows:
The Matthieu group $M_{24}$ is the automorphism group of the binary Golay code; this is a vector space over the field $\mathbb{F}_2$. The other groups can be obtained as stabilisers of various (sets of) elements of the Golay code, and hence are subgroups of $M_{24}$. The Mathieu groups form the so-called first generation of the happy family: the collection of 20 sporadic groups which are subgroups of the Monster group.
N-cafe blogpost on the groupoid $M_{13}$.
John Conway, Noam D. Elkies and Jeremy L. Martin, “The Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$”, Experimental Mathematics 15 (2006), 223–236. Eprint.
See also
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