In the area of modern algebra known as group theory, the Mathieu group M<sub>11</sub> is a sporadic simple group of order
M<sub>11</sub> is one of the 26 sporadic groups and was introduced by . It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial.
M<sub>11</sub> is a sharply 4-transitive permutation group on 11 objects. It admits many generating sets of permutations, such as the pair (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) of permutations used by the GAP computer algebra system.
M<sub>11</sub> has a sharply 4-transitive permutation representation on 11 points. The point stabilizer is sometimes denoted by M<sub>10</sub>, and is a non-split extension of the form A<sub>6</sub>.2 (an extension of the group of order 2 by the alternating group A<sub>6</sub>). This action is the automorphism group of a Steiner system S(4,5,11). The induced action on unordered pairs of points gives a rank 3 action on 55 points.
M<sub>11</sub> has a 3-transitive permutation representation on 12 points with point stabilizer PSL<sub>2</sub>(11). The permutation representations on 11 and 12 points can both be seen inside the Mathieu group M<sub>12</sub> as two different embeddings of M<sub>11</sub> in M<sub>12</sub>, exchanged by an outer automorphism.
The permutation representation on 11 points gives a complex irreducible representation in 10 dimensions. This is the smallest possible dimension of a faithful complex representation, though there are also two other such representations in 10 dimensions forming a complex conjugate pair.
M<sub>11</sub> has two 5-dimensional irreducible representations over the field with 3 elements, related to the restrictions of 6-dimensional representations of the double cover of M<sub>12</sub>. These have the smallest dimension of any faithful linear representations of M<sub>11</sub> over any field.
There are 5 conjugacy classes of maximal subgroups of M<sub>11</sub> as follows:
The maximum order of any element in M<sub>11</sub> is 11. Cycle structures are shown for the representations both of degree 11 and 12.