Symmetric representation of the elements of the Conway group ⋅0

In this paper we represent each element of the Conway group ⋅0 as a permutation on 24 letters from the Mathieu group M24, 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 , the sets of 24 mutually orthogonal 4-vectors in the Leech lattice Λ often referred to as frames of reference or crosses, as they are fundamental to our procedure. In particular we describe the 19 orbits of M24 on these crosses.