In
mathematics, the
Conway groups Co
1, Co
2, and Co
3 are three sporadic groups discovered by
John Horton Conway:
- Conway, J. H. A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups. Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 398-400.
All are closely related to the
Leech lattice Λ. The largest, Co
1 (of order 8,315,553,613,086,720,000), is obtained by dividing the
automorphism group of Λ by its
center, which consists of the scalar matrices ±1. The groups Co
2 (of order 42,305,421,312,000) and Co
3 (of order 495,766,656,000) consist of the automorphisms of Λ fixing a lattice vector of length 2 and a vector of √6 respectively. As the scalar -1 fixes no non-zero vector, we can regard these two groups as subgroups of Co
1.
Other sporadic groups
The groups Co2 and Co3 both contain the McLaughlin group McL (of order 898,128,000) and the Higman-Sims group (of order 44,352,000), which can be described as the pointwise stabilizers of a 2-2-√6 triangle and a 2-√6-√6 triangle respectively. Identifying R24 with C12 and Λ with Z[e2πi/3]12, the resulting auotmorphism group, i.e., the group of Leech lattice automorphisms preserving the complex structure, when divided by the 6-element group of complex scalar matrices, gives the Suzuki group Suz (of order 448,345,497,600). A similar construction gives the Hall-Janko group J2 (of order 604,800) as the quotient of the group of quaternionic automorphisms of Λ by the group ±1 of scalars.
See:
- Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press. ISBN 0-19-853199-0
