Mathieu group M24
M24 is the subgroup of S24 that is generated by the three permutations:[1] M24 can also be generated by two permutations:[2] 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.
A Fano subplane likewise satisfies suitable uniqueness conditions.
An S(3,6,22) system W22 is formed by appending just one new point to each of the 21 lines and new blocks are 56 hyperovals conjugate under M21.
Those remaining octads are subsets of W21 and are symmetric differences of pairs of lines.
Codewords correspond in a natural way to subsets of a set of 24 objects.
Those subsets corresponding to codewords with 8 or 12 coordinates equal to 1 are called octads or dodecads respectively.
This permutation can be described by starting with the tiling of the Klein quartic by 56 triangles (with 24 vertices – the 24 points on which the group acts), then forming squares of out some of the 2 triangles, and octagons out of 6 triangles, with the added permutation being "interchange the two endpoints of those edges of the original triangular tiling which bisect the squares and octagons".
[2] The theory of umbral moonshine is a partly conjectural relationship between K3 surfaces and M24.
)[citation needed] Frobenius (1904) calculated the complex character table of M24.
The subquotients give two irreducible representations of dimension 11 over the field with 2 elements.
[citation needed] Choi (1972b) found the 9 conjugacy classes of maximal subgroups of M24.
Curtis (1977) gave a short proof of the result, describing the 9 classes in terms of combinatorial data on the 24 points: the subgroups fix a point, duad, octad, duum, sextet, triad, trio, projective line, or octern, as described below.
Todd (1966) gave the character tables of M24 (originally calculated by Frobenius (1904)) and the 8 maximal subgroups that were known at the time.
197–208) A duum is a pair of complementary dodecads (12 point sets) in the Golay code.
The subgroup M12 acts differently on 2 sets of 12, reflecting the outer automorphism of M12.
26:(3.S6), order 138240: sextet group Consider a tetrad, any set of 4 points in the Steiner system W24.
The sextet group has a normal abelian subgroup H of order 64, isomorphic to the hexacode, a vector space of length 6 and dimension 3 over F4.
3.S6 is the normalizer in M24 of the subgroup generated by r=(BCD)(FGH)(JKL)(NOP)(RST)(VWX), which can be thought of as a multiplication of row numbers by u2.
The group 3.A6 is isomorphic to a subgroup of SL(3,4) whose image in PSL(3,4) has been noted[by whom?]
The subgroup fixing a triad is PSL(3,4):S3, order 120960, with orbits of size 3 and 21.
The subgroup fixing a projective line structure on the 24 points is PSL(2,23), order 6072, whose action is doubly transitive.
