The conjugate classes of the cubic surface group in an orthogonal representation

Abstract

The cubic surface group, of order 51840, has a representation by orthogonal matrices, of 5 rows and determinant + 1, over GF (3). It can be partitioned into conjugate classes on geometrical grounds because each matrix has two skew linear spaces, S + of even and S - of odd dimension, of latent points; the matrices fall into categories A, B, C according as the join of S + and S - has dimension 4, 2, 0. Subdivisions of A, B, C rest on the relation of S + and S - to the invariant quadric of the orthogonal group. A accounts for the identity matrix and the 4 types of involutions. B falls into two parts; one of 4 classes, discussed in §§5 to 8, the other of 9 classes, discussed in §§9 to 14. §§ 15 and 16 mention criteria for checking the number of operations in a conjugate class. Those classes in category C fall into 3 subcategories of 3, 2, 2 classes and are described in §§ 18 to 25.