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  "body": "\n## Description\n\nThe Monster group $\\mathbb{M}$ is the largest of the $26$ [[sporadic groups|PZ49Z4CBFZ]].\nIt is a finite [[simple group|0YMYMNQ0QZ]] of [[order|0B85S2V2DR]]\n$$\\abs{\\mathbb{M}} = 2^{46} \\cdot 3^{20} \\cdot 5^9 \\cdot 7^6 \\cdot 11^2 \\cdot 13^3 \\cdot 17 \\cdot 19 \\cdot 23 \\cdot 29 \\cdot 31 \\cdot 41 \\cdot 47 \\cdot 59 \\cdot 71,$$\nwhich is $808017424794512875886459904961710757005754368000000000$, about $8 \\cdot 10^{53}$.\n\n## Remarks\n\nThe smallest nontrivial [[irreducible representation|FX3NEFYFFN]] of $\\mathbb{M}$ over $\\C$ has dimension $196883$.\nRobert Griess constructed $\\mathbb{M}$ as the automorphism group of a commutative non-associative algebra of dimension $196884$, the [[Griess algebra|ZMPYZ8KD9N]].\nIts existence had been predicted earlier by Bernd Fischer and by Griess.\nThe [[$j$-invariant|FEC2XH822R]] has the expansion $j(\\tau) = q^{-1} + 744 + 196884 q + \\cdots$ with $q = e^{2 \\pi i \\tau}$, and $196884 = 196883 + 1$.\nThis coincidence led to the [[monstrous moonshine|E0Y7Y75T6C]] conjecture of Conway and Norton, proved by Richard Borcherds.\nTwenty of the $26$ sporadic groups are subquotients of $\\mathbb{M}$, that is, quotients of subgroups of $\\mathbb{M}$; the other six are the [[pariah groups|ZYZ5FYG7QC]].\n\n## Explanations\n\n### L1 human\n\nThe Monster is the largest of the exceptional finite simple groups, with about $8 \\cdot 10^{53}$ elements.\nIts smallest faithful action by complex matrices needs matrices of size $196883 \\times 196883$.\nIt contains most of the other exceptional simple groups, and it is linked in surprising ways to number theory through monstrous moonshine.\n",
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