Mathieu group M24
In the area of modern algebra known as group theory, the Mathieu group M24 is a sporadic simple group of order
M24 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer automorphism group are both trivial.
The Mathieu groups can be constructed in various ways. Initially, Mathieu and others constructed them as permutation groups. It was difficult to see that M24 actually existed, that its generators did not just generate the alternating group A24. The matter was clarified when Ernst Witt constructed M24 as the automorphism (symmetry) group of an S(5,8,24) Steiner system W24 (the Witt design). M24 is the group of permutations that map every block in this design to some other block. The subgroups M23 and M22 then are easily defined to be the stabilizers of a single point and a pair of points respectively.
M24 can be built starting from PSL(3,4), the projective special linear group of 3-dimensional space over the finite field with 4 elements (Dixon & Mortimer 1996, pp. 192–205), also called M21 which acts on the projective plane over the field F4, an S(2,5,21) system called W21. Its 21 blocks are called lines. Any 2 lines intersect at one point.
M21 has 168 simple subgroups of order 360 and 360 simple subgroups of order 168. In the larger projective general linear group PGL(3,4) both sets of subgroups form single conjugacy classes, but in M21 both sets split into 3 conjugacy classes. The subgroups respectively have orbits of 6, called hyperovals, and orbits of 7, called Fano subplanes. These sets allow creation of new blocks for larger Steiner systems. M21 is normal in PGL(3,4), of index 3. PGL(3,4) has an outer automorphism induced by transposing conjugate elements in F4 (the field automorphism). PGL(3,4) can therefore be extended to the group PΓL(3,4) of projective semilinear transformations, which is a split extension of M21 by the symmetric group S3. PΓL(3,4) has an embedding as a maximal subgroup of M24.(Griess 1998, p. 55])
...
Wikipedia