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GroupTheory

 ConwayGroup

 Calling Sequence ConwayGroup( n )

Parameters

 n - : {1,2,3} : integer parameter selecting the Conway group

Description

 • The Conway groups ${\mathrm{Co}}_{1}$, ${\mathrm{Co}}_{2}$ and ${\mathrm{Co}}_{3}$ are sporadic finite simple groups, discovered by John H. Conway in the 1960s.
 • The largest Conway group ${\mathrm{Co}}_{1}$ is the quotient of the automorphism group of the Leech lattice (a certain $24$-dimensional unimodular lattice) by its center (of order $2$), and has order equal to $4157776806543360000$.
 • The second Conway group ${\mathrm{Co}}_{2}$ has order equal to $42305421312000$, and is the group of automorphisms of the Leech lattice fixing a vector of length $2$.
 • The third Conway group ${\mathrm{Co}}_{3}$ has order equal to $495766656000$, and can be described as the group of automorphisms of the Leech lattice fixing a vector of length $\sqrt{6}$.
 • The ConwayGroup( n ) command returns a permutation group isomorphic to the Conway group ${\mathrm{Co}}_{n}$, for n = 1,2,3.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ConwayGroup}\left(2\right)$
 ${G}{:=}{{Co}}_{{2}}$ (1)
 > $\mathrm{Degree}\left(G\right)$
 ${2300}$ (2)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${42305421312000}$ (3)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (4)

Compatibility

 • The GroupTheory[ConwayGroup] command was introduced in Maple 17.