AbstractLet Ω be a set of 24 points with the structure of the (5,8,24) Steiner system,S , defined on it. The automorphism group of S acts on the famous Leech lattice,as does the binary Golay code defined by S . Let A, B ⊂ Ω be subsets of sizeour (“tetrads”). The structure of S forces each tetrad to define a certain partitionof Ω into six tetrads called a sextet. For each tetrad Conway defined a certainautomorphism of the Leech lattice that extends the group generated by the aboveo the full automorphism group of the lattice. For the tetrad A he denoted thisautomorphism ζA. It is well known that for ζA and ζB to commute it is sufficiento have A and B belong to the same sextet. We extend this to a much less obviousnecessary and sufficient condition, namely ζA and ζB will commute if and only ifA∪ B is contained in a block of S . We go on to...