metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D184, C23⋊1D8, C8⋊1D23, C184⋊1C2, D92⋊1C2, 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
Generators and relations for D184
G = < a,b | a184=b2=1, bab=a-1 >
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
Export