A330584 The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the automorphism groups of sublattices of the Leech lattice.

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 20160, 20160, 25920, 62400, 95040, 126000, 181440, 443520, 604800, 979200, 1451520, 1814400, 3265920, 4245696, 10200960

Offset: 1

Author

Hal M. Switkay, Dec 18 2019

Keywords

• nonn
• fini
• full

Comments

Note: not every sublattice of the Leech lattice is necessarily a section of the Leech lattice. For example, every Niemeyer lattice is commensurable with the Leech lattice; thus the orders of the simple components of their automorphism groups are in this list, even when those groups are not sections of Co0.

By a theorem of Conway and Sloane, any simple group with a cover that has a crystallographic representation in <= 21 dimensions is in this list.

This is a subsequence of A330583.

Examples

			All simple groups of order less than 9828 have crystallographic representations within sublattices of the Leech lattice. The smallest nontrivial crystallographic representation of L2(27), of order 9828, is 26-dimensional.
		

References

• J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
• J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer, 3rd ed., 1999.

Links

• Hal M. Switkay, Table of n, a(n) for n = 1..56
• J. H. Conway, N. J. A. Sloane, Low-dimensional lattices V: Integral coordinates for integral lattices, Proc. Royal Soc. A 426 (1989), 211-232.
• David A. Madore, Orders of non-abelian simple groups
• R. A. Wilson et al., ATLAS of Finite Group Representations - Version 3

Crossrefs

Cf. A109379, A080683, A330583.