Editing Mathieu group (section)

=== Automorphism groups on the Golay code ===
The group ''M''<sub>24</sub> is the [[Mathieu group M24#Automorphism group of the Golay code|permutation automorphism group]] of the [[binary Golay code|extended binary Golay code]] ''W'', i.e., the group of permutations on the 24 coordinates that map ''W'' to itself. All the Mathieu groups can be constructed as groups of permutations on the binary Golay code.

''M''<sub>12</sub> has index 2 in its automorphism group, and ''M''<sub>12</sub>:2 happens to be isomorphic to a subgroup of ''M''<sub>24</sub>. ''M''<sub>12</sub> is the stabilizer of a '''dodecad''', a codeword of 12 1's; ''M''<sub>12</sub>:2 stabilizes a partition into 2 complementary dodecads.

There is a natural connection between the Mathieu groups and the larger [[Conway groups]], because the [[Leech lattice]] was constructed on the binary Golay code and in fact both lie in spaces of dimension 24. The Conway groups in turn are found in the [[Monster group]].  [[Robert Griess]] refers to the 20 sporadic groups found in the Monster as the '''Happy Family''', and to the Mathieu groups as the '''first generation'''.