OFFSET
1,1
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.
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
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Hal M. Switkay, Dec 18 2019
STATUS
approved