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Steven H. Cullinane

Generating the Octad Generator

Expository note.  April 28, 1985.

The image  “MOGgen01.gif” cannot be displayed, because it contains errors.

S1 and S2 acting on row 1 below yield
the Miracle Octad Generator [3]:

The image “MOGCurtis03.gif” cannot be displayed, because it contains errors.

Apart from its use in studying the 759 octads of a Steiner system S(5,8,24) -- and hence the Mathieu group M24 -- the Curtis MOG nicely illustrates a natural correspondence C (Conwell [2], p. 72) between

(a)  the 35 partitions of an 8-set H (such as GF(8) above, or Conwell's 8 "heptads") into two 4-sets, and

(b)  the 35 partitions of L into four parallel affine planes.

Two of the H-partitions have a common refinement into 2-sets iff the same is true of the corresponding L-partitions.  (Cameron [1], p. 60).

Note that C is particularly natural in row 1, and that partitions 2-5 in each row have similar structures.

  1. Cameron, P. J., Parallelisms of Complete Designs, Camb. U. Pr. 1976.
  2. Conwell, G. M., The 3-space PG(3,2) and its group, Ann. of Math. 11 (1910) 60-76.
  3. Curtis, R. T., A new combinatorial approach to M24, Math. Proc. Camb. Phil. Soc. 79 (1976) 25-42.

For an image of the original 1985 typed note, click here.