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G = D169order 338 = 2·132

Dihedral group

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

C1C169 — D169
C1C13C169 — D169
C169 — D169
C1

Generators and relations for D169
 G = < a,b | a169=b2=1, bab=a-1 >

169C2
13D13

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···13F169A···169BZ
order1213···13169···169
size11692···22···2

86 irreducible representations

dim1122
type++++
imageC1C2D13D169
kernelD169C169C13C1
# reps11678

Matrix representation of D169 in GL2(𝔽677) generated by

33797
580179
,
33797
588340
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

Export

Subgroup lattice of D169 in TeX

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