Twelve Points in PG(5, 3) with 95040 Self-Transformations

H. S. M. Coxeter

Abstract

The title of this paper could have been 'Geometry in five dimensions over GF(3)' (cf. Edge 1954), or 'The geometry of the second Mathieu group', or 'Duads and synthemes', or 'Hexastigms', or simply 'Some thoughts on the number 6'. The words actually chosen acknowledge the inspiration of the late H. F. Baker, whose last book (Baker 1946) develops the idea of duads and synthemes in a different direction. The special property of the number 6 that makes the present development possible is the existence of an outer automorphism for the symmetric group of this degree. The consequent group of order 1440 is described abstractly in section 1, topologically in section 2, and geometrically in section section 3 to 7. The kernel of the geometrical discussion is in section 5, where the chords of a non-ruled quadric in the finite projective space PG(3, 3) are identified with the edges of a graph having an unusually high degree of regularity (Tutte 1958). It is seen in section 4 that the ten points which constitute this quadric can be derived very simply from a 'hexastigm' consisting of six points in PG(4, 3) (cf. Coxeter 1958). The connexion with Edge's work is described in section 6. Then section 7 shows that the derivation of the quadric from a hexastigm can be carried out in two distinct ways, suggesting the use of a second hexastigm in a different 4-space. It is found in section 8 that the consequent configuration of twelve points in PG(5, 3) can be divided into two hexastigms in 66 ways. The whole set of 132 hexastigms forms a geometrical realization of the Steiner system $\germ{S}$(5, 6, 12), whose group is known to be the quintuply transitive Mathieu group M$_{12}$, of order 95040. Finally, section 9 shows how the same 5-dimensional configuration can be regarded (in 396 ways) as a pair of mutually inscribed simplexes, like Mobius's mutually inscribed tetrahedra in ordinary space of 3 dimensions.