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  "body": "\n## Description\n\nThe Janko groups $J_1$, $J_2$, $J_3$ and $J_4$ are four [[sporadic groups|PZ49Z4CBFZ]], found by Zvonimir Janko.\nTheir [[orders|0B85S2V2DR]] are:\n\n1. $\\abs{J_1} = 175560 = 2^3 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot 11 \\cdot 19$;\n2. $\\abs{J_2} = 604800 = 2^7 \\cdot 3^3 \\cdot 5^2 \\cdot 7$;\n3. $\\abs{J_3} = 50232960 = 2^7 \\cdot 3^5 \\cdot 5 \\cdot 17 \\cdot 19$;\n4. $\\abs{J_4} = 86775571046077562880 = 2^{21} \\cdot 3^3 \\cdot 5 \\cdot 7 \\cdot 11^3 \\cdot 23 \\cdot 29 \\cdot 31 \\cdot 37 \\cdot 43$.\n\nEach of them is a finite [[simple group|0YMYMNQ0QZ]].\n\n## Remarks\n\n$J_1$ was the first sporadic group found after the [[Mathieu groups|T1KR4KPX8P]], about a century later.\nJanko characterised it by the [[centraliser|02YYV36BFM]] of an involution, which is $C_2 \\times A_5$.\n$J_1$ is a subgroup of the Chevalley group $G_2(\\F_{11})$.\n$J_2$ is also called the Hall–Janko group.\n$J_1$, $J_3$ and $J_4$ are [[pariah groups|ZYZ5FYG7QC]], since they are not subquotients of the [[Monster group|C9VY20TAPT]], while $J_2$ is a subquotient of the Monster.\n\n## Explanations\n\n### L1 human\n\nThe Janko groups are four of the exceptional finite simple groups.\nThe first of them showed, after a century without new examples, that the Mathieu groups were not the only exceptions.\n",
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