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  "body": "\n## Description\n\nThe Fischer groups $\\mathrm{Fi}_{22}$, $\\mathrm{Fi}_{23}$ and $\\mathrm{Fi}_{24}'$ are three [[sporadic groups|PZ49Z4CBFZ]], found by Bernd Fischer.\nTheir [[orders|0B85S2V2DR]] are:\n\n1. $\\abs{\\mathrm{Fi}_{22}} = 64561751654400 = 2^{17} \\cdot 3^9 \\cdot 5^2 \\cdot 7 \\cdot 11 \\cdot 13$;\n2. $\\abs{\\mathrm{Fi}_{23}} = 4089470473293004800 = 2^{18} \\cdot 3^{13} \\cdot 5^2 \\cdot 7 \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 23$;\n3. $\\abs{\\mathrm{Fi}_{24}'} = 1255205709190661721292800 = 2^{21} \\cdot 3^{16} \\cdot 5^2 \\cdot 7^3 \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 23 \\cdot 29$.\n\nEach of them is a finite [[simple group|0YMYMNQ0QZ]].\n\n## Remarks\n\nFischer found these groups while classifying 3-transposition groups.\nA 3-transposition group is a group generated by a [[conjugacy class|0DEBXAJ5SA]] $D$ of involutions such that the product of any two elements of $D$ has order $1$, $2$ or $3$.\nThe symmetric groups, with $D$ the class of transpositions, are the basic examples.\nThe group $\\mathrm{Fi}_{24}$ generated by Fischer's class of 3-transpositions is not simple; its [[commutator subgroup|0ASGSTXSBW]] $\\mathrm{Fi}_{24}'$ has index $2$ and is simple.\nAll three Fischer groups are subquotients of the [[Monster group|C9VY20TAPT]].\n\n## Explanations\n\n### L1 human\n\nIn a symmetric group, two transpositions multiply to an element of order $1$, $2$ or $3$.\nFischer asked which groups are generated by a class of elements of order $2$ with this property.\nBesides symmetric groups and some matrix groups, the answer contains three new simple groups, the Fischer groups.\n",
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