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  "body": "\n## Description\n\nThe Conway groups $\\mathrm{Co}_1$, $\\mathrm{Co}_2$ and $\\mathrm{Co}_3$ are three [[sporadic groups|PZ49Z4CBFZ]], found by John Conway from the symmetries of the [[Leech lattice|XETV3657DG]].\nLet $\\mathrm{Co}_0$ be the [[automorphism group|0K1SCPBNKJ]] of the Leech lattice, that is, the group of linear isometries of $\\R^{24}$ that map the lattice onto itself.\nIts [[centre|0DZ93493VN]] is $\\set{\\pm 1}$, and its order is $8315553613086720000$.\n\n1. $\\mathrm{Co}_1 = \\mathrm{Co}_0 / \\set{\\pm 1}$, of order $4157776806543360000 = 2^{21} \\cdot 3^9 \\cdot 5^4 \\cdot 7^2 \\cdot 11 \\cdot 13 \\cdot 23$.\n2. $\\mathrm{Co}_2$ is the stabiliser in $\\mathrm{Co}_0$ of a lattice vector of norm $4$, of order $42305421312000 = 2^{18} \\cdot 3^6 \\cdot 5^3 \\cdot 7 \\cdot 11 \\cdot 23$.\n3. $\\mathrm{Co}_3$ is the stabiliser in $\\mathrm{Co}_0$ of a lattice vector of norm $6$, of order $495766656000 = 2^{10} \\cdot 3^7 \\cdot 5^3 \\cdot 7 \\cdot 11 \\cdot 23$.\n\nAll three are [[simple groups|0YMYMNQ0QZ]].\nHere the lattice is scaled so that its shortest nonzero vectors have norm $4$.\n\n## Remarks\n\nThe group $\\mathrm{Co}_0$ itself is not simple, because of its centre $\\set{\\pm 1}$.\nSeveral other sporadic groups are subgroups of $\\mathrm{Co}_0$, among them the [[Mathieu group|T1KR4KPX8P]] $M_{24}$, the Higman–Sims group and the McLaughlin group.\nAll three Conway groups are subquotients of the [[Monster group|C9VY20TAPT]].\n\n## Explanations\n\n### L1 human\n\nThe Leech lattice is an extremely symmetric arrangement of points in $24$-dimensional space.\nIts symmetry group, divided by the symmetry $v \\mapsto -v$, is the simple group $\\mathrm{Co}_1$.\nThe groups $\\mathrm{Co}_2$ and $\\mathrm{Co}_3$ are the symmetries that keep one chosen lattice point fixed, for two kinds of points.\n",
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