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G = D164order 328 = 23·41

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D164, C4⋊D41, C411D4, C1641C2, D821C2, C2.4D82, C82.3C22, sometimes denoted D328 or Dih164 or Dih328, SmallGroup(328,6)

Series: Derived Chief Lower central Upper central

C1C82 — D164
C1C41C82D82 — D164
C41C82 — D164
C1C2C4

Generators and relations for D164
 G = < a,b | a164=b2=1, bab=a-1 >

82C2
82C2
41C22
41C22
2D41
2D41
41D4

Smallest permutation representation of D164
On 164 points
Generators in S164
(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)
(1 164)(2 163)(3 162)(4 161)(5 160)(6 159)(7 158)(8 157)(9 156)(10 155)(11 154)(12 153)(13 152)(14 151)(15 150)(16 149)(17 148)(18 147)(19 146)(20 145)(21 144)(22 143)(23 142)(24 141)(25 140)(26 139)(27 138)(28 137)(29 136)(30 135)(31 134)(32 133)(33 132)(34 131)(35 130)(36 129)(37 128)(38 127)(39 126)(40 125)(41 124)(42 123)(43 122)(44 121)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 105)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)

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

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

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

85 conjugacy classes

class 1 2A2B2C 4 41A···41T82A···82T164A···164AN
order1222441···4182···82164···164
size11828222···22···22···2

85 irreducible representations

dim1112222
type+++++++
imageC1C2C2D4D41D82D164
kernelD164C164D82C41C4C2C1
# reps1121202040

Matrix representation of D164 in GL2(𝔽821) generated by

674700
649685
,
353512
475468
G:=sub<GL(2,GF(821))| [674,649,700,685],[353,475,512,468] >;

D164 in GAP, Magma, Sage, TeX

D_{164}
% in TeX

G:=Group("D164");
// GroupNames label

G:=SmallGroup(328,6);
// by ID

G=gap.SmallGroup(328,6);
# by ID

G:=PCGroup([4,-2,-2,-2,-41,49,21,5123]);
// Polycyclic

G:=Group<a,b|a^164=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D164 in TeX

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