metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D169, C169⋊C2, C13.D13, sometimes denoted D338 or Dih169 or Dih338, SmallGroup(338,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C169 — D169 |
Generators and relations for D169
G = < a,b | a169=b2=1, bab=a-1 >
Smallest permutation representation of D169
►On 169 points
Generators in S169
(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)
(1 169)(2 168)(3 167)(4 166)(5 165)(6 164)(7 163)(8 162)(9 161)(10 160)(11 159)(12 158)(13 157)(14 156)(15 155)(16 154)(17 153)(18 152)(19 151)(20 150)(21 149)(22 148)(23 147)(24 146)(25 145)(26 144)(27 143)(28 142)(29 141)(30 140)(31 139)(32 138)(33 137)(34 136)(35 135)(36 134)(37 133)(38 132)(39 131)(40 130)(41 129)(42 128)(43 127)(44 126)(45 125)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)
G:=sub<Sym(169)| (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), (1,169)(2,168)(3,167)(4,166)(5,165)(6,164)(7,163)(8,162)(9,161)(10,160)(11,159)(12,158)(13,157)(14,156)(15,155)(16,154)(17,153)(18,152)(19,151)(20,150)(21,149)(22,148)(23,147)(24,146)(25,145)(26,144)(27,143)(28,142)(29,141)(30,140)(31,139)(32,138)(33,137)(34,136)(35,135)(36,134)(37,133)(38,132)(39,131)(40,130)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)>;
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), (1,169)(2,168)(3,167)(4,166)(5,165)(6,164)(7,163)(8,162)(9,161)(10,160)(11,159)(12,158)(13,157)(14,156)(15,155)(16,154)(17,153)(18,152)(19,151)(20,150)(21,149)(22,148)(23,147)(24,146)(25,145)(26,144)(27,143)(28,142)(29,141)(30,140)(31,139)(32,138)(33,137)(34,136)(35,135)(36,134)(37,133)(38,132)(39,131)(40,130)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86) );
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)], [(1,169),(2,168),(3,167),(4,166),(5,165),(6,164),(7,163),(8,162),(9,161),(10,160),(11,159),(12,158),(13,157),(14,156),(15,155),(16,154),(17,153),(18,152),(19,151),(20,150),(21,149),(22,148),(23,147),(24,146),(25,145),(26,144),(27,143),(28,142),(29,141),(30,140),(31,139),(32,138),(33,137),(34,136),(35,135),(36,134),(37,133),(38,132),(39,131),(40,130),(41,129),(42,128),(43,127),(44,126),(45,125),(46,124),(47,123),(48,122),(49,121),(50,120),(51,119),(52,118),(53,117),(54,116),(55,115),(56,114),(57,113),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86)]])
86 conjugacy classes
| class | 1 | 2 | 13A | ··· | 13F | 169A | ··· | 169BZ |
| order | 1 | 2 | 13 | ··· | 13 | 169 | ··· | 169 |
| size | 1 | 169 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 2 | 2 |
| type | + | + | + | + |
| image | C1 | C2 | D13 | D169 |
| kernel | D169 | C169 | C13 | C1 |
| # reps | 1 | 1 | 6 | 78 |
Matrix representation of D169 ►in GL2(𝔽677) generated by
| 337 | 97 |
| 580 | 179 |
| 337 | 97 |
| 588 | 340 |
G:=sub<GL(2,GF(677))| [337,580,97,179],[337,588,97,340] >;
D169 in GAP, Magma, Sage, TeX
D_{169} % in TeX
G:=Group("D169"); // GroupNames label
G:=SmallGroup(338,1);
// by ID
G=gap.SmallGroup(338,1);
# by ID
G:=PCGroup([3,-2,-13,-13,2017,82,2810]);
// Polycyclic
G:=Group<a,b|a^169=b^2=1,b*a*b=a^-1>;
// generators/relations
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