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G = D166order 332 = 22·83

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D166, C2×D83, C166⋊C2, C83⋊C22, sometimes denoted D332 or Dih166 or Dih332, SmallGroup(332,3)

Series: Derived Chief Lower central Upper central

C1C83 — D166
C1C83D83 — D166
C83 — D166
C1C2

Generators and relations for D166
 G = < a,b | a166=b2=1, bab=a-1 >

83C2
83C2
83C22

Smallest permutation representation of D166
On 166 points
Generators in S166
(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)
(1 166)(2 165)(3 164)(4 163)(5 162)(6 161)(7 160)(8 159)(9 158)(10 157)(11 156)(12 155)(13 154)(14 153)(15 152)(16 151)(17 150)(18 149)(19 148)(20 147)(21 146)(22 145)(23 144)(24 143)(25 142)(26 141)(27 140)(28 139)(29 138)(30 137)(31 136)(32 135)(33 134)(34 133)(35 132)(36 131)(37 130)(38 129)(39 128)(40 127)(41 126)(42 125)(43 124)(44 123)(45 122)(46 121)(47 120)(48 119)(49 118)(50 117)(51 116)(52 115)(53 114)(54 113)(55 112)(56 111)(57 110)(58 109)(59 108)(60 107)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)

G:=sub<Sym(166)| (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), (1,166)(2,165)(3,164)(4,163)(5,162)(6,161)(7,160)(8,159)(9,158)(10,157)(11,156)(12,155)(13,154)(14,153)(15,152)(16,151)(17,150)(18,149)(19,148)(20,147)(21,146)(22,145)(23,144)(24,143)(25,142)(26,141)(27,140)(28,139)(29,138)(30,137)(31,136)(32,135)(33,134)(34,133)(35,132)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,120)(48,119)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)>;

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), (1,166)(2,165)(3,164)(4,163)(5,162)(6,161)(7,160)(8,159)(9,158)(10,157)(11,156)(12,155)(13,154)(14,153)(15,152)(16,151)(17,150)(18,149)(19,148)(20,147)(21,146)(22,145)(23,144)(24,143)(25,142)(26,141)(27,140)(28,139)(29,138)(30,137)(31,136)(32,135)(33,134)(34,133)(35,132)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,120)(48,119)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84) );

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)], [(1,166),(2,165),(3,164),(4,163),(5,162),(6,161),(7,160),(8,159),(9,158),(10,157),(11,156),(12,155),(13,154),(14,153),(15,152),(16,151),(17,150),(18,149),(19,148),(20,147),(21,146),(22,145),(23,144),(24,143),(25,142),(26,141),(27,140),(28,139),(29,138),(30,137),(31,136),(32,135),(33,134),(34,133),(35,132),(36,131),(37,130),(38,129),(39,128),(40,127),(41,126),(42,125),(43,124),(44,123),(45,122),(46,121),(47,120),(48,119),(49,118),(50,117),(51,116),(52,115),(53,114),(54,113),(55,112),(56,111),(57,110),(58,109),(59,108),(60,107),(61,106),(62,105),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,98),(70,97),(71,96),(72,95),(73,94),(74,93),(75,92),(76,91),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84)])

86 conjugacy classes

class 1 2A2B2C83A···83AO166A···166AO
order122283···83166···166
size1183832···22···2

86 irreducible representations

dim11122
type+++++
imageC1C2C2D83D166
kernelD166D83C166C2C1
# reps1214141

Matrix representation of D166 in GL3(𝔽167) generated by

16600
0730
095121
,
100
04935
03118
G:=sub<GL(3,GF(167))| [166,0,0,0,7,95,0,30,121],[1,0,0,0,49,3,0,35,118] >;

D166 in GAP, Magma, Sage, TeX

D_{166}
% in TeX

G:=Group("D166");
// GroupNames label

G:=SmallGroup(332,3);
// by ID

G=gap.SmallGroup(332,3);
# by ID

G:=PCGroup([3,-2,-2,-83,2954]);
// Polycyclic

G:=Group<a,b|a^166=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D166 in TeX

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