The Coxeter – Todd lattice , the Mitchell group , and related sphere packings

This paper studies the Coxeter-Todd lattice Ag, its automorphism group (which is Mitchell's reflection group 6-P$C7(4,3)-2), and the associated 12-dimensional real lattice K12. We give several constructions for Ag, which is a Z[w]-lattice where w = e ; enumerate the congruence classes of Ag/2Ag and A%/6A%, where 6 = to — <D; prove the lattice is unique; determine its covering radius and deep holes; and study its connections with the lattice E6 and the Leech lattice. A number of new dense lattices in dimensions up to about 10 are constructed. We also give an explicit basis for the invariants of the Mitchell group. The paper concludes with an extensive bibliography.