Copied to
clipboard

## G = D183order 366 = 2·3·61

### Dihedral group

Aliases: D183, C61⋊S3, C3⋊D61, C1831C2, sometimes denoted D366 or Dih183 or Dih366, SmallGroup(366,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C183 — D183
 Chief series C1 — C61 — C183 — D183
 Lower central C183 — D183
 Upper central C1

Generators and relations for D183
G = < a,b | a183=b2=1, bab=a-1 >

183C2
61S3
3D61

Smallest permutation representation of D183
On 183 points
Generators in S183
```(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)
(1 183)(2 182)(3 181)(4 180)(5 179)(6 178)(7 177)(8 176)(9 175)(10 174)(11 173)(12 172)(13 171)(14 170)(15 169)(16 168)(17 167)(18 166)(19 165)(20 164)(21 163)(22 162)(23 161)(24 160)(25 159)(26 158)(27 157)(28 156)(29 155)(30 154)(31 153)(32 152)(33 151)(34 150)(35 149)(36 148)(37 147)(38 146)(39 145)(40 144)(41 143)(42 142)(43 141)(44 140)(45 139)(46 138)(47 137)(48 136)(49 135)(50 134)(51 133)(52 132)(53 131)(54 130)(55 129)(56 128)(57 127)(58 126)(59 125)(60 124)(61 123)(62 122)(63 121)(64 120)(65 119)(66 118)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)```

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

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

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

93 conjugacy classes

 class 1 2 3 61A ··· 61AD 183A ··· 183BH order 1 2 3 61 ··· 61 183 ··· 183 size 1 183 2 2 ··· 2 2 ··· 2

93 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 S3 D61 D183 kernel D183 C183 C61 C3 C1 # reps 1 1 1 30 60

Matrix representation of D183 in GL2(𝔽367) generated by

 208 14 64 119
,
 336 123 28 31
`G:=sub<GL(2,GF(367))| [208,64,14,119],[336,28,123,31] >;`

D183 in GAP, Magma, Sage, TeX

`D_{183}`
`% in TeX`

`G:=Group("D183");`
`// GroupNames label`

`G:=SmallGroup(366,5);`
`// by ID`

`G=gap.SmallGroup(366,5);`
`# by ID`

`G:=PCGroup([3,-2,-3,-61,25,3242]);`
`// Polycyclic`

`G:=Group<a,b|a^183=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

Export

׿
×
𝔽