Symmetric representation of the elements of the Conway group

In this paper we represent each element of the Conway group .0 as a permutation on 24 letters from the Mathieu group M-24, followed by a codeword of the binary Golay code (which corresponds to a diagonal matrix taking the value -1 on the positions of the codeword and 1 otherwise), followed by a word of length at most 4 in a highly symmetric generating set. We describe an algorithm for multiplying elements represented in this way, that we have implemented in MAGMA. We include a detailed description Of (Lambda) over bar (4), the sets of 24 mutually orthogonal 4-vectors in the Leech lattice Lambda often referred to as frames of reference or crosses, as they are fundamental to our procedure. In particular we describe the 19 orbits Of M-24 on these crosses. (C) 2009 Elsevier Ltd. All rights reserved.