Editing Conway group (section)

==Two other sporadic groups==
Two sporadic subgroups can be defined as quotients of stabilizers of structures on the Leech lattice. Identifying '''R'''<sup>24</sup> with '''C'''<sup>12</sup> and Λ with

: <math>\mathbf{Z}\left[e^{\frac{2}{3}\pi i}\right]^{12},</math>

the resulting automorphism group (i.e., the group of Leech lattice automorphisms preserving the [[complex structure on a real vector space|complex structure]]) when divided by the six-element group of complex scalar matrices, gives the '''[[Suzuki sporadic group|Suzuki group]]''' Suz (order {{val|fmt=commas|448,345,497,600}}). This group was discovered by [[Michio Suzuki (mathematician)|Michio Suzuki]] in 1968.

A similar construction gives the '''[[Hall–Janko group]]''' J<sub>2</sub> (order {{val|fmt=commas|604,800}}) as the quotient of the group of [[quaternion]]ic automorphisms of Λ by the group ±1 of scalars.

The seven simple groups described above comprise what [[Robert Griess]] calls the ''second generation of the Happy Family'', which consists of the 20 sporadic simple groups found within the [[Monster group]]. Several of the seven groups contain at least some of the five [[Mathieu groups]], which comprise the ''first generation''.