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## G = D184order 368 = 24·23

### Dihedral group

Aliases: D184, C231D8, C81D23, C1841C2, D921C2, C4.9D46, C2.4D92, C46.2D4, C92.9C22, sometimes denoted D368 or Dih184 or Dih368, SmallGroup(368,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C92 — D184
 Chief series C1 — C23 — C46 — C92 — D92 — D184
 Lower central C23 — C46 — C92 — D184
 Upper central C1 — C2 — C4 — C8

Generators and relations for D184
G = < a,b | a184=b2=1, bab=a-1 >

92C2
92C2
46C22
46C22
4D23
4D23
23D4
23D4
2D46
2D46
23D8

Smallest permutation representation of D184
On 184 points
Generators in S184
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184)
(1 184)(2 183)(3 182)(4 181)(5 180)(6 179)(7 178)(8 177)(9 176)(10 175)(11 174)(12 173)(13 172)(14 171)(15 170)(16 169)(17 168)(18 167)(19 166)(20 165)(21 164)(22 163)(23 162)(24 161)(25 160)(26 159)(27 158)(28 157)(29 156)(30 155)(31 154)(32 153)(33 152)(34 151)(35 150)(36 149)(37 148)(38 147)(39 146)(40 145)(41 144)(42 143)(43 142)(44 141)(45 140)(46 139)(47 138)(48 137)(49 136)(50 135)(51 134)(52 133)(53 132)(54 131)(55 130)(56 129)(57 128)(58 127)(59 126)(60 125)(61 124)(62 123)(63 122)(64 121)(65 120)(66 119)(67 118)(68 117)(69 116)(70 115)(71 114)(72 113)(73 112)(74 111)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)```

`G:=sub<Sym(184)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184), (1,184)(2,183)(3,182)(4,181)(5,180)(6,179)(7,178)(8,177)(9,176)(10,175)(11,174)(12,173)(13,172)(14,171)(15,170)(16,169)(17,168)(18,167)(19,166)(20,165)(21,164)(22,163)(23,162)(24,161)(25,160)(26,159)(27,158)(28,157)(29,156)(30,155)(31,154)(32,153)(33,152)(34,151)(35,150)(36,149)(37,148)(38,147)(39,146)(40,145)(41,144)(42,143)(43,142)(44,141)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,132)(54,131)(55,130)(56,129)(57,128)(58,127)(59,126)(60,125)(61,124)(62,123)(63,122)(64,121)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184), (1,184)(2,183)(3,182)(4,181)(5,180)(6,179)(7,178)(8,177)(9,176)(10,175)(11,174)(12,173)(13,172)(14,171)(15,170)(16,169)(17,168)(18,167)(19,166)(20,165)(21,164)(22,163)(23,162)(24,161)(25,160)(26,159)(27,158)(28,157)(29,156)(30,155)(31,154)(32,153)(33,152)(34,151)(35,150)(36,149)(37,148)(38,147)(39,146)(40,145)(41,144)(42,143)(43,142)(44,141)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,132)(54,131)(55,130)(56,129)(57,128)(58,127)(59,126)(60,125)(61,124)(62,123)(63,122)(64,121)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184)], [(1,184),(2,183),(3,182),(4,181),(5,180),(6,179),(7,178),(8,177),(9,176),(10,175),(11,174),(12,173),(13,172),(14,171),(15,170),(16,169),(17,168),(18,167),(19,166),(20,165),(21,164),(22,163),(23,162),(24,161),(25,160),(26,159),(27,158),(28,157),(29,156),(30,155),(31,154),(32,153),(33,152),(34,151),(35,150),(36,149),(37,148),(38,147),(39,146),(40,145),(41,144),(42,143),(43,142),(44,141),(45,140),(46,139),(47,138),(48,137),(49,136),(50,135),(51,134),(52,133),(53,132),(54,131),(55,130),(56,129),(57,128),(58,127),(59,126),(60,125),(61,124),(62,123),(63,122),(64,121),(65,120),(66,119),(67,118),(68,117),(69,116),(70,115),(71,114),(72,113),(73,112),(74,111),(75,110),(76,109),(77,108),(78,107),(79,106),(80,105),(81,104),(82,103),(83,102),(84,101),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93)])`

95 conjugacy classes

 class 1 2A 2B 2C 4 8A 8B 23A ··· 23K 46A ··· 46K 92A ··· 92V 184A ··· 184AR order 1 2 2 2 4 8 8 23 ··· 23 46 ··· 46 92 ··· 92 184 ··· 184 size 1 1 92 92 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

95 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 D4 D8 D23 D46 D92 D184 kernel D184 C184 D92 C46 C23 C8 C4 C2 C1 # reps 1 1 2 1 2 11 11 22 44

Matrix representation of D184 in GL2(𝔽1289) generated by

 1182 397 715 1178
,
 1102 790 1248 187
`G:=sub<GL(2,GF(1289))| [1182,715,397,1178],[1102,1248,790,187] >;`

D184 in GAP, Magma, Sage, TeX

`D_{184}`
`% in TeX`

`G:=Group("D184");`
`// GroupNames label`

`G:=SmallGroup(368,6);`
`// by ID`

`G=gap.SmallGroup(368,6);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-23,61,66,182,42,8804]);`
`// Polycyclic`

`G:=Group<a,b|a^184=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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