The Leech Lattice Lambda and the Conway group .O revisited

We give a new, concise definition of the Conway group .O as follows. The Mathieu group M-24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of ((24)(4)) tetrads. We use this action to define a progenitor P of shape 2*((24)(4)) : M-24; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P by this relator results in .O. Consideration of the lowest dimension in which .O can act faithfully produces Conway's elements xi(T) and the 24-dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under .O of the integral vectors in R-24.