Coxeter-Todd Lattice

The complex Lattice $\Lambda_6^\omega$ corresponding to real lattice $K_{12}$ having the densest Hypersphere Packing (Kissing Number) in 12-D. The associated Automorphism Group $G_0$ was discovered by Mitchell (1914). The order of $G_0$ is given by

$$\vert\mathop{\rm Aut}\nolimits (\Lambda_6^\omega)\vert=2^9\cdot 3^7\cdot 5\cdot 7=39,191,040.$$

The order of the Automorphism Group of $K_{12}$ is given by

$$\vert\mathop{\rm Aut}\nolimits (K_{12})\vert=2^{10}\cdot 3^7\cdot 5\cdot 7$$

(Conway and Sloane 1983).

See also Barnes-Wall Lattice, Leech Lattice

References

Conway, J. H. and Sloane, N. J. A. ``The Coxeter-Todd Lattice, the Mitchell Group and Related Sphere Packings.'' Math. Proc. Camb. Phil. Soc. 93, 421-440, 1983.

Conway, J. H. and Sloane, N. J. A. ``The 12-Dimensional Coxeter-Todd Lattice $K_{12}$.'' §4.9 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 127-129, 1993.

Coxeter, H. S. M. and Todd, J. A. ``As Extreme Duodenary Form.'' Canad. J. Math. 5, 384-392, 1953.

Mitchell, H. H. ``Determination of All Primitive Collineation Groups in More than Four Variables.'' Amer. J. Math. 36, 1-12, 1914.

Todd, J. A. ``The Characters of a Collineation Group in Five Dimensions.'' Proc. Roy. Soc. London Ser. A 200, 320-336, 1950.