Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
Symmetric representation of the elements of the Janko group J1
Journal of Symbolic Computation
| Hi-index | 0.00 |
In this paper we represent each element of the Conway group @?0 as a permutation on 24 letters from the Mathieu group M"2"4, 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 @L@?"4, the sets of 24 mutually orthogonal 4-vectors in the Leech lattice @L 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"2"4 on these crosses.