Mathieu group M22

In the area of modern algebra known as group theory, the Mathieu group M₂₂ is a sporadic simple group of order

M₂₂ is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M₂₂ is cyclic of order 12, and the outer automorphism group has order 2.

There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature. Burgoyne & Fong (1966) incorrectly claimed that the Schur multiplier of M₂₂ has order 3, and in a correction Burgoyne & Fong (1968) incorrectly claimed that it has order 6. This caused an error in the title of the paper Janko (1976) announcing the discovery of the Janko group J4. Mazet (1979) showed that the Schur multiplier is in fact cyclic of order 12.

Adem & Milgram (1995) calculated the 2-part of all the cohomology of M₂₂.

M₂₂ has a 3-transitive permutation representation on 22 points, with point stabilizer the group PSL₃(4), sometimes called M₂₁. This action fixes a Steiner system S(3,6,22) with 77 hexads, whose full automorphism group is the automorphism group M₂₂.2 of M₂₂.

M₂₂ has three rank 3 permutation representations: one on the 77 hexads with point stabilizer 2⁴:A₆, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A₇.

M₂₂ is the point stabilizer of the action of M₂₃ on 23 points, and also the point stabilizer of the rank 3 action of the Higman–Sims group on 100 = 1+22+77 points.

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