metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D161, C23⋊D7, C7⋊D23, C161⋊1C2, sometimes denoted D322 or Dih161 or Dih322, SmallGroup(322,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C161 — D161 |
Generators and relations for D161
G = < a,b | a161=b2=1, bab=a-1 >
Smallest permutation representation of D161
►On 161 points
Generators in S161
(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)
(1 161)(2 160)(3 159)(4 158)(5 157)(6 156)(7 155)(8 154)(9 153)(10 152)(11 151)(12 150)(13 149)(14 148)(15 147)(16 146)(17 145)(18 144)(19 143)(20 142)(21 141)(22 140)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(41 121)(42 120)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 113)(50 112)(51 111)(52 110)(53 109)(54 108)(55 107)(56 106)(57 105)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)
G:=sub<Sym(161)| (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), (1,161)(2,160)(3,159)(4,158)(5,157)(6,156)(7,155)(8,154)(9,153)(10,152)(11,151)(12,150)(13,149)(14,148)(15,147)(16,146)(17,145)(18,144)(19,143)(20,142)(21,141)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)>;
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), (1,161)(2,160)(3,159)(4,158)(5,157)(6,156)(7,155)(8,154)(9,153)(10,152)(11,151)(12,150)(13,149)(14,148)(15,147)(16,146)(17,145)(18,144)(19,143)(20,142)(21,141)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82) );
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)], [(1,161),(2,160),(3,159),(4,158),(5,157),(6,156),(7,155),(8,154),(9,153),(10,152),(11,151),(12,150),(13,149),(14,148),(15,147),(16,146),(17,145),(18,144),(19,143),(20,142),(21,141),(22,140),(23,139),(24,138),(25,137),(26,136),(27,135),(28,134),(29,133),(30,132),(31,131),(32,130),(33,129),(34,128),(35,127),(36,126),(37,125),(38,124),(39,123),(40,122),(41,121),(42,120),(43,119),(44,118),(45,117),(46,116),(47,115),(48,114),(49,113),(50,112),(51,111),(52,110),(53,109),(54,108),(55,107),(56,106),(57,105),(58,104),(59,103),(60,102),(61,101),(62,100),(63,99),(64,98),(65,97),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82)]])
82 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 23A | ··· | 23K | 161A | ··· | 161BN |
| order | 1 | 2 | 7 | 7 | 7 | 23 | ··· | 23 | 161 | ··· | 161 |
| size | 1 | 161 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D7 | D23 | D161 |
| kernel | D161 | C161 | C23 | C7 | C1 |
| # reps | 1 | 1 | 3 | 11 | 66 |
Matrix representation of D161 ►in GL2(𝔽967) generated by
| 11 | 905 |
| 268 | 951 |
| 248 | 707 |
| 575 | 719 |
G:=sub<GL(2,GF(967))| [11,268,905,951],[248,575,707,719] >;
D161 in GAP, Magma, Sage, TeX
D_{161} % in TeX
G:=Group("D161"); // GroupNames label
G:=SmallGroup(322,3);
// by ID
G=gap.SmallGroup(322,3);
# by ID
G:=PCGroup([3,-2,-7,-23,73,2774]);
// Polycyclic
G:=Group<a,b|a^161=b^2=1,b*a*b=a^-1>;
// generators/relations
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