direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D170, C2×D85, C34⋊D5, C10⋊D17, C5⋊2D34, C17⋊2D10, C170⋊1C2, C85⋊2C22, sometimes denoted D340 or Dih170 or Dih340, SmallGroup(340,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C85 — D170 |
Generators and relations for D170
G = < a,b | a170=b2=1, bab=a-1 >
Smallest permutation representation of D170
►On 170 points
Generators in S170
(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)
(1 170)(2 169)(3 168)(4 167)(5 166)(6 165)(7 164)(8 163)(9 162)(10 161)(11 160)(12 159)(13 158)(14 157)(15 156)(16 155)(17 154)(18 153)(19 152)(20 151)(21 150)(22 149)(23 148)(24 147)(25 146)(26 145)(27 144)(28 143)(29 142)(30 141)(31 140)(32 139)(33 138)(34 137)(35 136)(36 135)(37 134)(38 133)(39 132)(40 131)(41 130)(42 129)(43 128)(44 127)(45 126)(46 125)(47 124)(48 123)(49 122)(50 121)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 90)(82 89)(83 88)(84 87)(85 86)
G:=sub<Sym(170)| (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), (1,170)(2,169)(3,168)(4,167)(5,166)(6,165)(7,164)(8,163)(9,162)(10,161)(11,160)(12,159)(13,158)(14,157)(15,156)(16,155)(17,154)(18,153)(19,152)(20,151)(21,150)(22,149)(23,148)(24,147)(25,146)(26,145)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170), (1,170)(2,169)(3,168)(4,167)(5,166)(6,165)(7,164)(8,163)(9,162)(10,161)(11,160)(12,159)(13,158)(14,157)(15,156)(16,155)(17,154)(18,153)(19,152)(20,151)(21,150)(22,149)(23,148)(24,147)(25,146)(26,145)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170)], [(1,170),(2,169),(3,168),(4,167),(5,166),(6,165),(7,164),(8,163),(9,162),(10,161),(11,160),(12,159),(13,158),(14,157),(15,156),(16,155),(17,154),(18,153),(19,152),(20,151),(21,150),(22,149),(23,148),(24,147),(25,146),(26,145),(27,144),(28,143),(29,142),(30,141),(31,140),(32,139),(33,138),(34,137),(35,136),(36,135),(37,134),(38,133),(39,132),(40,131),(41,130),(42,129),(43,128),(44,127),(45,126),(46,125),(47,124),(48,123),(49,122),(50,121),(51,120),(52,119),(53,118),(54,117),(55,116),(56,115),(57,114),(58,113),(59,112),(60,111),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(81,90),(82,89),(83,88),(84,87),(85,86)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 17A | ··· | 17H | 34A | ··· | 34H | 85A | ··· | 85AF | 170A | ··· | 170AF |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 17 | ··· | 17 | 34 | ··· | 34 | 85 | ··· | 85 | 170 | ··· | 170 |
| size | 1 | 1 | 85 | 85 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D5 | D10 | D17 | D34 | D85 | D170 |
| kernel | D170 | D85 | C170 | C34 | C17 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 8 | 32 | 32 |
Matrix representation of D170 ►in GL3(𝔽1021) generated by
| 1020 | 0 | 0 |
| 0 | 813 | 595 |
| 0 | 426 | 92 |
| 1 | 0 | 0 |
| 0 | 813 | 595 |
| 0 | 799 | 208 |
G:=sub<GL(3,GF(1021))| [1020,0,0,0,813,426,0,595,92],[1,0,0,0,813,799,0,595,208] >;
D170 in GAP, Magma, Sage, TeX
D_{170} % in TeX
G:=Group("D170"); // GroupNames label
G:=SmallGroup(340,14);
// by ID
G=gap.SmallGroup(340,14);
# by ID
G:=PCGroup([4,-2,-2,-5,-17,194,5123]);
// Polycyclic
G:=Group<a,b|a^170=b^2=1,b*a*b=a^-1>;
// generators/relations
Export