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G = D167order 334 = 2·167

Dihedral group

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

Aliases: D167, C167⋊C2, sometimes denoted D334 or Dih167 or Dih334, SmallGroup(334,1)

Series: Derived Chief Lower central Upper central

C1C167 — D167
C1C167 — D167
C167 — D167
C1

Generators and relations for D167
 G = < a,b | a167=b2=1, bab=a-1 >

167C2

Smallest permutation representation of D167
On 167 points: primitive
Generators in S167
(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)
(1 167)(2 166)(3 165)(4 164)(5 163)(6 162)(7 161)(8 160)(9 159)(10 158)(11 157)(12 156)(13 155)(14 154)(15 153)(16 152)(17 151)(18 150)(19 149)(20 148)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 140)(29 139)(30 138)(31 137)(32 136)(33 135)(34 134)(35 133)(36 132)(37 131)(38 130)(39 129)(40 128)(41 127)(42 126)(43 125)(44 124)(45 123)(46 122)(47 121)(48 120)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 108)(61 107)(62 106)(63 105)(64 104)(65 103)(66 102)(67 101)(68 100)(69 99)(70 98)(71 97)(72 96)(73 95)(74 94)(75 93)(76 92)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)

G:=sub<Sym(167)| (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), (1,167)(2,166)(3,165)(4,164)(5,163)(6,162)(7,161)(8,160)(9,159)(10,158)(11,157)(12,156)(13,155)(14,154)(15,153)(16,152)(17,151)(18,150)(19,149)(20,148)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,133)(36,132)(37,131)(38,130)(39,129)(40,128)(41,127)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,108)(61,107)(62,106)(63,105)(64,104)(65,103)(66,102)(67,101)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)>;

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), (1,167)(2,166)(3,165)(4,164)(5,163)(6,162)(7,161)(8,160)(9,159)(10,158)(11,157)(12,156)(13,155)(14,154)(15,153)(16,152)(17,151)(18,150)(19,149)(20,148)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,133)(36,132)(37,131)(38,130)(39,129)(40,128)(41,127)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,108)(61,107)(62,106)(63,105)(64,104)(65,103)(66,102)(67,101)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85) );

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)], [(1,167),(2,166),(3,165),(4,164),(5,163),(6,162),(7,161),(8,160),(9,159),(10,158),(11,157),(12,156),(13,155),(14,154),(15,153),(16,152),(17,151),(18,150),(19,149),(20,148),(21,147),(22,146),(23,145),(24,144),(25,143),(26,142),(27,141),(28,140),(29,139),(30,138),(31,137),(32,136),(33,135),(34,134),(35,133),(36,132),(37,131),(38,130),(39,129),(40,128),(41,127),(42,126),(43,125),(44,124),(45,123),(46,122),(47,121),(48,120),(49,119),(50,118),(51,117),(52,116),(53,115),(54,114),(55,113),(56,112),(57,111),(58,110),(59,109),(60,108),(61,107),(62,106),(63,105),(64,104),(65,103),(66,102),(67,101),(68,100),(69,99),(70,98),(71,97),(72,96),(73,95),(74,94),(75,93),(76,92),(77,91),(78,90),(79,89),(80,88),(81,87),(82,86),(83,85)])

85 conjugacy classes

class 1  2 167A···167CE
order12167···167
size11672···2

85 irreducible representations

dim112
type+++
imageC1C2D167
kernelD167C167C1
# reps1183

Matrix representation of D167 in GL2(𝔽2339) generated by

10342338
10
,
10342338
2321305
G:=sub<GL(2,GF(2339))| [1034,1,2338,0],[1034,232,2338,1305] >;

D167 in GAP, Magma, Sage, TeX

D_{167}
% in TeX

G:=Group("D167");
// GroupNames label

G:=SmallGroup(334,1);
// by ID

G=gap.SmallGroup(334,1);
# by ID

G:=PCGroup([2,-2,-167,1329]);
// Polycyclic

G:=Group<a,b|a^167=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D167 in TeX

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