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## G = D163order 326 = 2·163

### Dihedral group

Aliases: D163, C163⋊C2, sometimes denoted D326 or Dih163 or Dih326, SmallGroup(326,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C163 — D163
 Chief series C1 — C163 — D163
 Lower central C163 — D163
 Upper central C1

Generators and relations for D163
G = < a,b | a163=b2=1, bab=a-1 >

163C2

Smallest permutation representation of D163
On 163 points: primitive
Generators in S163
```(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)
(1 163)(2 162)(3 161)(4 160)(5 159)(6 158)(7 157)(8 156)(9 155)(10 154)(11 153)(12 152)(13 151)(14 150)(15 149)(16 148)(17 147)(18 146)(19 145)(20 144)(21 143)(22 142)(23 141)(24 140)(25 139)(26 138)(27 137)(28 136)(29 135)(30 134)(31 133)(32 132)(33 131)(34 130)(35 129)(36 128)(37 127)(38 126)(39 125)(40 124)(41 123)(42 122)(43 121)(44 120)(45 119)(46 118)(47 117)(48 116)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 108)(57 107)(58 106)(59 105)(60 104)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)```

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

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

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

83 conjugacy classes

 class 1 2 163A ··· 163CC order 1 2 163 ··· 163 size 1 163 2 ··· 2

83 irreducible representations

 dim 1 1 2 type + + + image C1 C2 D163 kernel D163 C163 C1 # reps 1 1 81

Matrix representation of D163 in GL2(𝔽653) generated by

 611 652 1 0
,
 611 652 457 42
`G:=sub<GL(2,GF(653))| [611,1,652,0],[611,457,652,42] >;`

D163 in GAP, Magma, Sage, TeX

`D_{163}`
`% in TeX`

`G:=Group("D163");`
`// GroupNames label`

`G:=SmallGroup(326,1);`
`// by ID`

`G=gap.SmallGroup(326,1);`
`# by ID`

`G:=PCGroup([2,-2,-163,1297]);`
`// Polycyclic`

`G:=Group<a,b|a^163=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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