Mathieu group M22
In the area of modern algebra known as group theory, the Mathieu group "M" is a sporadic simple group of order History and properties. "M" is one of the 26 sporadic groups and was introduced by . It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M 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. Representations. M has a 3-transitive permutation representation on 22 points, with point stabilizer the  group PSL(4), sometimes called M. This action fixes a Steiner system S(3,6,22) with 77 hexads, whose full automorphism group is the automorphism group M.2 of M. M has three rank 3 permutation representations: one on the 77 hexads with point stabilizer 2:A, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A. M is the point stabilizer of the action of M on 23 points, and also the point stabilizer of the rank 3 action of the Higman–Sims group on 100 = 1+22+77 points . The triple cover 3.M has a 6-dimensional faithful representation over the field with 4 elements. The 6-fold cover of M appears in the centralizer 2.3.(M:2) of an involution of the Janko group J4. Maximal subgroups. There are no proper subgroups transitive on all 22 points. There are 8 conjugacy classes of maximal subgroups of "M" as follows: Conjugacy classes. There are 12 conjugacy classes, though the two classes of elements of order 11 are fused under an outer automorphism.