direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D182, C2×D91, C26⋊D7, C14⋊D13, C7⋊2D26, C13⋊2D14, C182⋊1C2, C91⋊2C22, sometimes denoted D364 or Dih182 or Dih364, SmallGroup(364,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C91 — D182 |
Generators and relations for D182
G = < a,b | a182=b2=1, bab=a-1 >
Smallest permutation representation of D182
►On 182 points
Generators in S182
(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)
(1 182)(2 181)(3 180)(4 179)(5 178)(6 177)(7 176)(8 175)(9 174)(10 173)(11 172)(12 171)(13 170)(14 169)(15 168)(16 167)(17 166)(18 165)(19 164)(20 163)(21 162)(22 161)(23 160)(24 159)(25 158)(26 157)(27 156)(28 155)(29 154)(30 153)(31 152)(32 151)(33 150)(34 149)(35 148)(36 147)(37 146)(38 145)(39 144)(40 143)(41 142)(42 141)(43 140)(44 139)(45 138)(46 137)(47 136)(48 135)(49 134)(50 133)(51 132)(52 131)(53 130)(54 129)(55 128)(56 127)(57 126)(58 125)(59 124)(60 123)(61 122)(62 121)(63 120)(64 119)(65 118)(66 117)(67 116)(68 115)(69 114)(70 113)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)
G:=sub<Sym(182)| (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), (1,182)(2,181)(3,180)(4,179)(5,178)(6,177)(7,176)(8,175)(9,174)(10,173)(11,172)(12,171)(13,170)(14,169)(15,168)(16,167)(17,166)(18,165)(19,164)(20,163)(21,162)(22,161)(23,160)(24,159)(25,158)(26,157)(27,156)(28,155)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,140)(44,139)(45,138)(46,137)(47,136)(48,135)(49,134)(50,133)(51,132)(52,131)(53,130)(54,129)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)>;
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), (1,182)(2,181)(3,180)(4,179)(5,178)(6,177)(7,176)(8,175)(9,174)(10,173)(11,172)(12,171)(13,170)(14,169)(15,168)(16,167)(17,166)(18,165)(19,164)(20,163)(21,162)(22,161)(23,160)(24,159)(25,158)(26,157)(27,156)(28,155)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,140)(44,139)(45,138)(46,137)(47,136)(48,135)(49,134)(50,133)(51,132)(52,131)(53,130)(54,129)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92) );
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)], [(1,182),(2,181),(3,180),(4,179),(5,178),(6,177),(7,176),(8,175),(9,174),(10,173),(11,172),(12,171),(13,170),(14,169),(15,168),(16,167),(17,166),(18,165),(19,164),(20,163),(21,162),(22,161),(23,160),(24,159),(25,158),(26,157),(27,156),(28,155),(29,154),(30,153),(31,152),(32,151),(33,150),(34,149),(35,148),(36,147),(37,146),(38,145),(39,144),(40,143),(41,142),(42,141),(43,140),(44,139),(45,138),(46,137),(47,136),(48,135),(49,134),(50,133),(51,132),(52,131),(53,130),(54,129),(55,128),(56,127),(57,126),(58,125),(59,124),(60,123),(61,122),(62,121),(63,120),(64,119),(65,118),(66,117),(67,116),(68,115),(69,114),(70,113),(71,112),(72,111),(73,110),(74,109),(75,108),(76,107),(77,106),(78,105),(79,104),(80,103),(81,102),(82,101),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | 7B | 7C | 13A | ··· | 13F | 14A | 14B | 14C | 26A | ··· | 26F | 91A | ··· | 91AJ | 182A | ··· | 182AJ |
| order | 1 | 2 | 2 | 2 | 7 | 7 | 7 | 13 | ··· | 13 | 14 | 14 | 14 | 26 | ··· | 26 | 91 | ··· | 91 | 182 | ··· | 182 |
| size | 1 | 1 | 91 | 91 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
94 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D7 | D13 | D14 | D26 | D91 | D182 |
| kernel | D182 | D91 | C182 | C26 | C14 | C13 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 3 | 6 | 3 | 6 | 36 | 36 |
Matrix representation of D182 ►in GL3(𝔽547) generated by
| 546 | 0 | 0 |
| 0 | 83 | 236 |
| 0 | 311 | 133 |
| 1 | 0 | 0 |
| 0 | 194 | 467 |
| 0 | 26 | 353 |
G:=sub<GL(3,GF(547))| [546,0,0,0,83,311,0,236,133],[1,0,0,0,194,26,0,467,353] >;
D182 in GAP, Magma, Sage, TeX
D_{182} % in TeX
G:=Group("D182"); // GroupNames label
G:=SmallGroup(364,10);
// by ID
G=gap.SmallGroup(364,10);
# by ID
G:=PCGroup([4,-2,-2,-7,-13,290,5379]);
// Polycyclic
G:=Group<a,b|a^182=b^2=1,b*a*b=a^-1>;
// generators/relations
Export