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## G = D169order 338 = 2·132

### Dihedral group

Aliases: D169, C169⋊C2, C13.D13, sometimes denoted D338 or Dih169 or Dih338, SmallGroup(338,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C169 — D169
 Chief series C1 — C13 — C169 — D169
 Lower central C169 — D169
 Upper central C1

Generators and relations for D169
G = < a,b | a169=b2=1, bab=a-1 >

169C2
13D13

Smallest permutation representation of D169
On 169 points
Generators in S169
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169)
(1 169)(2 168)(3 167)(4 166)(5 165)(6 164)(7 163)(8 162)(9 161)(10 160)(11 159)(12 158)(13 157)(14 156)(15 155)(16 154)(17 153)(18 152)(19 151)(20 150)(21 149)(22 148)(23 147)(24 146)(25 145)(26 144)(27 143)(28 142)(29 141)(30 140)(31 139)(32 138)(33 137)(34 136)(35 135)(36 134)(37 133)(38 132)(39 131)(40 130)(41 129)(42 128)(43 127)(44 126)(45 125)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)```

`G:=sub<Sym(169)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169), (1,169)(2,168)(3,167)(4,166)(5,165)(6,164)(7,163)(8,162)(9,161)(10,160)(11,159)(12,158)(13,157)(14,156)(15,155)(16,154)(17,153)(18,152)(19,151)(20,150)(21,149)(22,148)(23,147)(24,146)(25,145)(26,144)(27,143)(28,142)(29,141)(30,140)(31,139)(32,138)(33,137)(34,136)(35,135)(36,134)(37,133)(38,132)(39,131)(40,130)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169), (1,169)(2,168)(3,167)(4,166)(5,165)(6,164)(7,163)(8,162)(9,161)(10,160)(11,159)(12,158)(13,157)(14,156)(15,155)(16,154)(17,153)(18,152)(19,151)(20,150)(21,149)(22,148)(23,147)(24,146)(25,145)(26,144)(27,143)(28,142)(29,141)(30,140)(31,139)(32,138)(33,137)(34,136)(35,135)(36,134)(37,133)(38,132)(39,131)(40,130)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169)], [(1,169),(2,168),(3,167),(4,166),(5,165),(6,164),(7,163),(8,162),(9,161),(10,160),(11,159),(12,158),(13,157),(14,156),(15,155),(16,154),(17,153),(18,152),(19,151),(20,150),(21,149),(22,148),(23,147),(24,146),(25,145),(26,144),(27,143),(28,142),(29,141),(30,140),(31,139),(32,138),(33,137),(34,136),(35,135),(36,134),(37,133),(38,132),(39,131),(40,130),(41,129),(42,128),(43,127),(44,126),(45,125),(46,124),(47,123),(48,122),(49,121),(50,120),(51,119),(52,118),(53,117),(54,116),(55,115),(56,114),(57,113),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86)]])`

86 conjugacy classes

 class 1 2 13A ··· 13F 169A ··· 169BZ order 1 2 13 ··· 13 169 ··· 169 size 1 169 2 ··· 2 2 ··· 2

86 irreducible representations

 dim 1 1 2 2 type + + + + image C1 C2 D13 D169 kernel D169 C169 C13 C1 # reps 1 1 6 78

Matrix representation of D169 in GL2(𝔽677) generated by

 337 97 580 179
,
 337 97 588 340
`G:=sub<GL(2,GF(677))| [337,580,97,179],[337,588,97,340] >;`

D169 in GAP, Magma, Sage, TeX

`D_{169}`
`% in TeX`

`G:=Group("D169");`
`// GroupNames label`

`G:=SmallGroup(338,1);`
`// by ID`

`G=gap.SmallGroup(338,1);`
`# by ID`

`G:=PCGroup([3,-2,-13,-13,2017,82,2810]);`
`// Polycyclic`

`G:=Group<a,b|a^169=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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