THE LEECH LATTICE Λ AND THE CONWAY GROUP .O REVISITED

We give a new, concise definition of the Conway group ·O as follows.The Mathieu group M24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of $(_4^{24} )$tetrads. We use this action to define a progenitor P of shape $2*(_4^{24} )$M24; 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 elementsοT and the 24— dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under of the integral vectors in R24.