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G = D173order 346 = 2·173

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D173, C173⋊C2, sometimes denoted D346 or Dih173 or Dih346, SmallGroup(346,1)

Series: Derived Chief Lower central Upper central

C1C173 — D173
C1C173 — D173
C173 — D173
C1

Generators and relations for D173
 G = < a,b | a173=b2=1, bab=a-1 >

173C2

Smallest permutation representation of D173
On 173 points: primitive
Generators in S173
(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)
(1 173)(2 172)(3 171)(4 170)(5 169)(6 168)(7 167)(8 166)(9 165)(10 164)(11 163)(12 162)(13 161)(14 160)(15 159)(16 158)(17 157)(18 156)(19 155)(20 154)(21 153)(22 152)(23 151)(24 150)(25 149)(26 148)(27 147)(28 146)(29 145)(30 144)(31 143)(32 142)(33 141)(34 140)(35 139)(36 138)(37 137)(38 136)(39 135)(40 134)(41 133)(42 132)(43 131)(44 130)(45 129)(46 128)(47 127)(48 126)(49 125)(50 124)(51 123)(52 122)(53 121)(54 120)(55 119)(56 118)(57 117)(58 116)(59 115)(60 114)(61 113)(62 112)(63 111)(64 110)(65 109)(66 108)(67 107)(68 106)(69 105)(70 104)(71 103)(72 102)(73 101)(74 100)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)

G:=sub<Sym(173)| (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), (1,173)(2,172)(3,171)(4,170)(5,169)(6,168)(7,167)(8,166)(9,165)(10,164)(11,163)(12,162)(13,161)(14,160)(15,159)(16,158)(17,157)(18,156)(19,155)(20,154)(21,153)(22,152)(23,151)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(31,143)(32,142)(33,141)(34,140)(35,139)(36,138)(37,137)(38,136)(39,135)(40,134)(41,133)(42,132)(43,131)(44,130)(45,129)(46,128)(47,127)(48,126)(49,125)(50,124)(51,123)(52,122)(53,121)(54,120)(55,119)(56,118)(57,117)(58,116)(59,115)(60,114)(61,113)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)>;

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), (1,173)(2,172)(3,171)(4,170)(5,169)(6,168)(7,167)(8,166)(9,165)(10,164)(11,163)(12,162)(13,161)(14,160)(15,159)(16,158)(17,157)(18,156)(19,155)(20,154)(21,153)(22,152)(23,151)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(31,143)(32,142)(33,141)(34,140)(35,139)(36,138)(37,137)(38,136)(39,135)(40,134)(41,133)(42,132)(43,131)(44,130)(45,129)(46,128)(47,127)(48,126)(49,125)(50,124)(51,123)(52,122)(53,121)(54,120)(55,119)(56,118)(57,117)(58,116)(59,115)(60,114)(61,113)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88) );

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)], [(1,173),(2,172),(3,171),(4,170),(5,169),(6,168),(7,167),(8,166),(9,165),(10,164),(11,163),(12,162),(13,161),(14,160),(15,159),(16,158),(17,157),(18,156),(19,155),(20,154),(21,153),(22,152),(23,151),(24,150),(25,149),(26,148),(27,147),(28,146),(29,145),(30,144),(31,143),(32,142),(33,141),(34,140),(35,139),(36,138),(37,137),(38,136),(39,135),(40,134),(41,133),(42,132),(43,131),(44,130),(45,129),(46,128),(47,127),(48,126),(49,125),(50,124),(51,123),(52,122),(53,121),(54,120),(55,119),(56,118),(57,117),(58,116),(59,115),(60,114),(61,113),(62,112),(63,111),(64,110),(65,109),(66,108),(67,107),(68,106),(69,105),(70,104),(71,103),(72,102),(73,101),(74,100),(75,99),(76,98),(77,97),(78,96),(79,95),(80,94),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88)])

88 conjugacy classes

class 1  2 173A···173CH
order12173···173
size11732···2

88 irreducible representations

dim112
type+++
imageC1C2D173
kernelD173C173C1
# reps1186

Matrix representation of D173 in GL2(𝔽347) generated by

31346
22360
,
15648
281191
G:=sub<GL(2,GF(347))| [31,223,346,60],[156,281,48,191] >;

D173 in GAP, Magma, Sage, TeX

D_{173}
% in TeX

G:=Group("D173");
// GroupNames label

G:=SmallGroup(346,1);
// by ID

G=gap.SmallGroup(346,1);
# by ID

G:=PCGroup([2,-2,-173,1377]);
// Polycyclic

G:=Group<a,b|a^173=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D173 in TeX

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