The simple group Co1 of order is defined as the quotient of Co0 by its center, which consists of the scalar matrices ±1.
It is common to speak of the type of a Leech lattice vector: half the square norm.
Thomas Thompson (1983) relates how, in about 1964, John Leech investigated close packings of spheres in Euclidean spaces of large dimension.
Conway expected to spend months or years on the problem, but found results in just a few sessions.
Witt (1998, page 329) stated that he found the Leech lattice in 1940 and hinted that he calculated the order of its automorphism group Co0.
A standard representation, used throughout this article, of the binary Golay code arranges the 24 co-ordinates so that 6 consecutive blocks (tetrads) of 4 constitute a sextet.
The Leech lattice can easily be defined as the Z-module generated by the set Λ2 of all vectors of type 2, consisting of and their images under N. Λ2 under N falls into 3 orbits of sizes 1104, 97152, and 98304.
Since Λ includes vectors of the shape (±8, 023), Co0 consists of rational matrices whose denominators are all divisors of 8.
Conway then named stabilizers of planes defined by triangles having the origin as a vertex.
In the simplest cases Co0 is transitive on the points or triangles in question and stabilizer groups are defined up to conjugacy.
Here is a table[3][4] of some sublattice groups: Two sporadic subgroups can be defined as quotients of stabilizers of structures on the Leech lattice.
In M24 an element of shape 38 generates a group normal in a copy of S3, which commutes with a simple subgroup of order 168.
In Co0 this monomial normalizer 24:PSL(2,7) × S3 is expanded to a maximal subgroup of the form 2.A9 × S3, where 2.A9 is the double cover of the alternating group A9.
John Thompson pointed out it would be fruitful to investigate the normalizers of smaller subgroups of the form 2.An (Conway 1971, p. 242).
The chain ends with 6.Suz:2 (Suz=Suzuki sporadic group), which, as mentioned above, respects a complex representation of the Leech Lattice.
Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups.
= {1, 0, 276, 2,048, 11,202, 49,152, ...} (OEIS: A097340) where one can set the constant term a(0) = 24, and η(τ) is the Dedekind eta function.