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## G = D151order 302 = 2·151

### Dihedral group

Aliases: D151, C151⋊C2, sometimes denoted D302 or Dih151 or Dih302, SmallGroup(302,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C151 — D151
 Chief series C1 — C151 — D151
 Lower central C151 — D151
 Upper central C1

Generators and relations for D151
G = < a,b | a151=b2=1, bab=a-1 >

151C2

Smallest permutation representation of D151
On 151 points: primitive
Generators in S151
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151)
(1 151)(2 150)(3 149)(4 148)(5 147)(6 146)(7 145)(8 144)(9 143)(10 142)(11 141)(12 140)(13 139)(14 138)(15 137)(16 136)(17 135)(18 134)(19 133)(20 132)(21 131)(22 130)(23 129)(24 128)(25 127)(26 126)(27 125)(28 124)(29 123)(30 122)(31 121)(32 120)(33 119)(34 118)(35 117)(36 116)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 100)(53 99)(54 98)(55 97)(56 96)(57 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)```

`G:=sub<Sym(151)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,144)(9,143)(10,142)(11,141)(12,140)(13,139)(14,138)(15,137)(16,136)(17,135)(18,134)(19,133)(20,132)(21,131)(22,130)(23,129)(24,128)(25,127)(26,126)(27,125)(28,124)(29,123)(30,122)(31,121)(32,120)(33,119)(34,118)(35,117)(36,116)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,98)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,144)(9,143)(10,142)(11,141)(12,140)(13,139)(14,138)(15,137)(16,136)(17,135)(18,134)(19,133)(20,132)(21,131)(22,130)(23,129)(24,128)(25,127)(26,126)(27,125)(28,124)(29,123)(30,122)(31,121)(32,120)(33,119)(34,118)(35,117)(36,116)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,98)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151)], [(1,151),(2,150),(3,149),(4,148),(5,147),(6,146),(7,145),(8,144),(9,143),(10,142),(11,141),(12,140),(13,139),(14,138),(15,137),(16,136),(17,135),(18,134),(19,133),(20,132),(21,131),(22,130),(23,129),(24,128),(25,127),(26,126),(27,125),(28,124),(29,123),(30,122),(31,121),(32,120),(33,119),(34,118),(35,117),(36,116),(37,115),(38,114),(39,113),(40,112),(41,111),(42,110),(43,109),(44,108),(45,107),(46,106),(47,105),(48,104),(49,103),(50,102),(51,101),(52,100),(53,99),(54,98),(55,97),(56,96),(57,95),(58,94),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77)]])`

77 conjugacy classes

 class 1 2 151A ··· 151BW order 1 2 151 ··· 151 size 1 151 2 ··· 2

77 irreducible representations

 dim 1 1 2 type + + + image C1 C2 D151 kernel D151 C151 C1 # reps 1 1 75

Matrix representation of D151 in GL2(𝔽907) generated by

 431 906 1 0
,
 431 906 732 476
`G:=sub<GL(2,GF(907))| [431,1,906,0],[431,732,906,476] >;`

D151 in GAP, Magma, Sage, TeX

`D_{151}`
`% in TeX`

`G:=Group("D151");`
`// GroupNames label`

`G:=SmallGroup(302,1);`
`// by ID`

`G=gap.SmallGroup(302,1);`
`# by ID`

`G:=PCGroup([2,-2,-151,1201]);`
`// Polycyclic`

`G:=Group<a,b|a^151=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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