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G = D163order 326 = 2·163

Dihedral group

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

Aliases: D163, C163⋊C2, sometimes denoted D326 or Dih163 or Dih326, SmallGroup(326,1)

Series: Derived Chief Lower central Upper central

C1C163 — D163
C1C163 — D163
C163 — D163
C1

Generators and relations for D163
 G = < a,b | a163=b2=1, bab=a-1 >

163C2

Smallest permutation representation of D163
On 163 points: primitive
Generators in S163
(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)
(1 163)(2 162)(3 161)(4 160)(5 159)(6 158)(7 157)(8 156)(9 155)(10 154)(11 153)(12 152)(13 151)(14 150)(15 149)(16 148)(17 147)(18 146)(19 145)(20 144)(21 143)(22 142)(23 141)(24 140)(25 139)(26 138)(27 137)(28 136)(29 135)(30 134)(31 133)(32 132)(33 131)(34 130)(35 129)(36 128)(37 127)(38 126)(39 125)(40 124)(41 123)(42 122)(43 121)(44 120)(45 119)(46 118)(47 117)(48 116)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 108)(57 107)(58 106)(59 105)(60 104)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)

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

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

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

83 conjugacy classes

class 1  2 163A···163CC
order12163···163
size11632···2

83 irreducible representations

dim112
type+++
imageC1C2D163
kernelD163C163C1
# reps1181

Matrix representation of D163 in GL2(𝔽653) generated by

611652
10
,
611652
45742
G:=sub<GL(2,GF(653))| [611,1,652,0],[611,457,652,42] >;

D163 in GAP, Magma, Sage, TeX

D_{163}
% in TeX

G:=Group("D163");
// GroupNames label

G:=SmallGroup(326,1);
// by ID

G=gap.SmallGroup(326,1);
# by ID

G:=PCGroup([2,-2,-163,1297]);
// Polycyclic

G:=Group<a,b|a^163=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D163 in TeX

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