Exceptional Object - Unimodular Lattices

Unimodular Lattices

Up to isometry there is only one even unimodular lattice in 15 dimensions or less — the E₈ lattice. Up to dimension 24 there is only one even unimodular lattice with no roots, the Leech lattice. Three of the sporadic simple groups were discovered by Conway while investigating the automorphism group of the Leech lattice. For example Co₁ is the automorphism group itself modulo ±1. The groups Co₂ and Co₃, as well as a number of other sporadic groups, arise as stabilisers of various subsets of the Leech lattice.