The Leech lattice $\Lambda$ and the Conway group $\cdot {O}$ revisited

We give a new, concise definition of the Conway group $\cdot {O}$ as follows. The Mathieu group $\mathrm{M}_{24}$ acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of $\left({24}\atop{4} \right)$ tetrads. We use this action to define a progenitor $P$ of shape $2^{\star \left( 24 \atop 4 \right)}:\mathrm{M}_{24}$; 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 $\cdot {O}$. Consideration of the lowest dimension in which $\cdot {O}$ 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 $\cdot {O}$ of the integral vectors in ${\mathbb{R}}_{24}$.