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## G = D142order 284 = 22·71

### Dihedral group

Aliases: D142, C2×D71, C142⋊C2, C71⋊C22, sometimes denoted D284 or Dih142 or Dih284, SmallGroup(284,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C71 — D142
 Chief series C1 — C71 — D71 — D142
 Lower central C71 — D142
 Upper central C1 — C2

Generators and relations for D142
G = < a,b | a142=b2=1, bab=a-1 >

Smallest permutation representation of D142
On 142 points
Generators in S142
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142)
(1 142)(2 141)(3 140)(4 139)(5 138)(6 137)(7 136)(8 135)(9 134)(10 133)(11 132)(12 131)(13 130)(14 129)(15 128)(16 127)(17 126)(18 125)(19 124)(20 123)(21 122)(22 121)(23 120)(24 119)(25 118)(26 117)(27 116)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 104)(40 103)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 96)(48 95)(49 94)(50 93)(51 92)(52 91)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)```

`G:=sub<Sym(142)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142), (1,142)(2,141)(3,140)(4,139)(5,138)(6,137)(7,136)(8,135)(9,134)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142), (1,142)(2,141)(3,140)(4,139)(5,138)(6,137)(7,136)(8,135)(9,134)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142)], [(1,142),(2,141),(3,140),(4,139),(5,138),(6,137),(7,136),(8,135),(9,134),(10,133),(11,132),(12,131),(13,130),(14,129),(15,128),(16,127),(17,126),(18,125),(19,124),(20,123),(21,122),(22,121),(23,120),(24,119),(25,118),(26,117),(27,116),(28,115),(29,114),(30,113),(31,112),(32,111),(33,110),(34,109),(35,108),(36,107),(37,106),(38,105),(39,104),(40,103),(41,102),(42,101),(43,100),(44,99),(45,98),(46,97),(47,96),(48,95),(49,94),(50,93),(51,92),(52,91),(53,90),(54,89),(55,88),(56,87),(57,86),(58,85),(59,84),(60,83),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72)])`

74 conjugacy classes

 class 1 2A 2B 2C 71A ··· 71AI 142A ··· 142AI order 1 2 2 2 71 ··· 71 142 ··· 142 size 1 1 71 71 2 ··· 2 2 ··· 2

74 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D71 D142 kernel D142 D71 C142 C2 C1 # reps 1 2 1 35 35

Matrix representation of D142 in GL3(𝔽569) generated by

 568 0 0 0 236 176 0 236 0
,
 1 0 0 0 116 412 0 430 453
`G:=sub<GL(3,GF(569))| [568,0,0,0,236,236,0,176,0],[1,0,0,0,116,430,0,412,453] >;`

D142 in GAP, Magma, Sage, TeX

`D_{142}`
`% in TeX`

`G:=Group("D142");`
`// GroupNames label`

`G:=SmallGroup(284,3);`
`// by ID`

`G=gap.SmallGroup(284,3);`
`# by ID`

`G:=PCGroup([3,-2,-2,-71,2522]);`
`// Polycyclic`

`G:=Group<a,b|a^142=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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