Editing Mathieu group (section)

=== Automorphism groups of Steiner systems ===

There exists [[up to]] [[Equivalence relation|equivalence]] a unique ''S''(5,8,24) [[Steiner system]] '''W<sub>24</sub>''' (the [[Witt design]]). The group ''M''<sub>24</sub> is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups ''M''<sub>23</sub> and ''M''<sub>22</sub> are defined to be the stabilizers of a single point and two points respectively.

Similarly, there exists up to equivalence a unique ''S''(5,6,12) Steiner system '''W<sub>12</sub>''', and the group ''M''<sub>12</sub> is its automorphism group. The subgroup ''M''<sub>11</sub> is the stabilizer of a point.

''W''<sub>12</sub> can be constructed from the [[affine geometry]] on the [[vector space]] {{nowrap|'''F'''<sub>3</sub> × '''F'''<sub>3</sub>}}, an ''S''(2,3,9) system.

An alternative construction of ''W''<sub>12</sub> is the "Kitten" of {{Harvtxt|Curtis|1984}}.

An introduction to a construction of ''W''<sub>24</sub> via the [[Miracle Octad Generator]] of R. T. Curtis and Conway's analog for ''W''<sub>12</sub>, the miniMOG, can be found in the book by Conway and [[Neil Sloane|Sloane]].