Symmetric representation of the elements of the Conway group \(.0\). (Q1023269)

The authors represent each element of the Conway group \(\cdot 0\) 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 position of the codeword and 1 otherwise), followed by a word of length at most 4 in a highly symmetric generating set. They describe an algorithm for multiplying elements represented in this way, that they have implemented in \textsc{Magma}. They include a detailed description of \(\overline\Lambda_4\), the sets of 24 mutually orthogonal 4-vectors in the Leech structure \(\Lambda\) often referred to as frames of reference or crosses, as they are fundamental to this procedure. In particular, the 19 orbits of \(M_{24}\) on these crosses are described.