direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D162, C2×D81, C162⋊C2, C81⋊C22, C3.D54, C27.D6, C54.2S3, C6.2D27, C9.1D18, C18.2D9, sometimes denoted D324 or Dih162 or Dih324, SmallGroup(324,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C81 — D162 |
Generators and relations for D162
G = < a,b | a162=b2=1, bab=a-1 >
Smallest permutation representation of D162
►On 162 points
Generators in S162
(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)
(1 162)(2 161)(3 160)(4 159)(5 158)(6 157)(7 156)(8 155)(9 154)(10 153)(11 152)(12 151)(13 150)(14 149)(15 148)(16 147)(17 146)(18 145)(19 144)(20 143)(21 142)(22 141)(23 140)(24 139)(25 138)(26 137)(27 136)(28 135)(29 134)(30 133)(31 132)(32 131)(33 130)(34 129)(35 128)(36 127)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 108)(56 107)(57 106)(58 105)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)
G:=sub<Sym(162)| (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), (1,162)(2,161)(3,160)(4,159)(5,158)(6,157)(7,156)(8,155)(9,154)(10,153)(11,152)(12,151)(13,150)(14,149)(15,148)(16,147)(17,146)(18,145)(19,144)(20,143)(21,142)(22,141)(23,140)(24,139)(25,138)(26,137)(27,136)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)>;
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), (1,162)(2,161)(3,160)(4,159)(5,158)(6,157)(7,156)(8,155)(9,154)(10,153)(11,152)(12,151)(13,150)(14,149)(15,148)(16,147)(17,146)(18,145)(19,144)(20,143)(21,142)(22,141)(23,140)(24,139)(25,138)(26,137)(27,136)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82) );
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)], [(1,162),(2,161),(3,160),(4,159),(5,158),(6,157),(7,156),(8,155),(9,154),(10,153),(11,152),(12,151),(13,150),(14,149),(15,148),(16,147),(17,146),(18,145),(19,144),(20,143),(21,142),(22,141),(23,140),(24,139),(25,138),(26,137),(27,136),(28,135),(29,134),(30,133),(31,132),(32,131),(33,130),(34,129),(35,128),(36,127),(37,126),(38,125),(39,124),(40,123),(41,122),(42,121),(43,120),(44,119),(45,118),(46,117),(47,116),(48,115),(49,114),(50,113),(51,112),(52,111),(53,110),(54,109),(55,108),(56,107),(57,106),(58,105),(59,104),(60,103),(61,102),(62,101),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(81,82)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 9A | 9B | 9C | 18A | 18B | 18C | 27A | ··· | 27I | 54A | ··· | 54I | 81A | ··· | 81AA | 162A | ··· | 162AA |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 9 | 9 | 9 | 18 | 18 | 18 | 27 | ··· | 27 | 54 | ··· | 54 | 81 | ··· | 81 | 162 | ··· | 162 |
| size | 1 | 1 | 81 | 81 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D9 | D18 | D27 | D54 | D81 | D162 |
| kernel | D162 | D81 | C162 | C54 | C27 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 3 | 3 | 9 | 9 | 27 | 27 |
Matrix representation of D162 ►in GL3(𝔽163) generated by
| 162 | 0 | 0 |
| 0 | 110 | 16 |
| 0 | 147 | 94 |
| 1 | 0 | 0 |
| 0 | 110 | 16 |
| 0 | 69 | 53 |
G:=sub<GL(3,GF(163))| [162,0,0,0,110,147,0,16,94],[1,0,0,0,110,69,0,16,53] >;
D162 in GAP, Magma, Sage, TeX
D_{162} % in TeX
G:=Group("D162"); // GroupNames label
G:=SmallGroup(324,4);
// by ID
G=gap.SmallGroup(324,4);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,362,284,1443,381,5404,208,7781]);
// Polycyclic
G:=Group<a,b|a^162=b^2=1,b*a*b=a^-1>;
// generators/relations
Export