metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D183, C61⋊S3, C3⋊D61, C183⋊1C2, sometimes denoted D366 or Dih183 or Dih366, SmallGroup(366,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C183 — D183 |
Generators and relations for D183
G = < a,b | a183=b2=1, bab=a-1 >
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
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