A second note on the simple group of order 6048

Abstract

1. The simple group of order 6048 has a permutation representation of degree 28 and two of degree 63. These are conspicuous in SU(3,32), the representation as a group of 3-rowed matrices, unitary and of determinant + 1, whose elements all belong to GF(32), or to J as we name this field. When these matrices as linear transformations in the projective plane ϖ over J they leave invariant the unit Hermitian formand the single capital U will henceforth signify the group so acting. There are 28 isotropic points n in ϖ satisfying H3 = 0; they furnish the representation of degree 28. The other 63 points p in ϖ are non-isotropic and furnish one of the representations of degree 63. The other is furnished by 63 triangles T; the vertices of a T are all p and every pair of them is linked in the antipolarity Π set up by H3; if their coordinate vectors are y and z then y′z = 0. Each p is a vertex of 3 T. These matters are fully explained in (7). Those points linked by II to a given point A are collinear, on the polar of A; no p lies on its polar, every n does. The polars t of points n are called tangents of H3 = 0, and the only isotropic point on t is its pole. The polars s of points p are secants of H3 = 0: each consists of 4 n with 3 pairs of linked p and, given s, it is these 3 pairs that complete those T having a vertex at the pole of s.