# Conway group

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}.

The groups Co_{2} and Co_{3} 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 **R**^{24} with **C**^{12} and Λ with **Z**[*e*^{2πi/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** J_{2} (of order 604,800) as the quotient of the group of quaternionic automorphisms of Λ by the group ±1 of scalars.

See: