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G = D162order 324 = 22·34

Dihedral group

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

Aliases: D162, C2×D81, C162⋊C2, C81⋊C22, C3.D54, C27.D6, C54.2S3, C6.2D27, C9.1D18, C18.2D9, sometimes denoted D324 or Dih162 or Dih324, SmallGroup(324,4)

Series: Derived Chief Lower central Upper central

C1C81 — D162
C1C3C9C27C81D81 — D162
C81 — D162
C1C2

Generators and relations for D162
 G = < a,b | a162=b2=1, bab=a-1 >

81C2
81C2
81C22
27S3
27S3
27D6
9D9
9D9
9D18
3D27
3D27
3D54

Smallest permutation representation of D162
On 162 points
Generators in S162
(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)
(1 162)(2 161)(3 160)(4 159)(5 158)(6 157)(7 156)(8 155)(9 154)(10 153)(11 152)(12 151)(13 150)(14 149)(15 148)(16 147)(17 146)(18 145)(19 144)(20 143)(21 142)(22 141)(23 140)(24 139)(25 138)(26 137)(27 136)(28 135)(29 134)(30 133)(31 132)(32 131)(33 130)(34 129)(35 128)(36 127)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 108)(56 107)(57 106)(58 105)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)

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

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

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

84 conjugacy classes

class 1 2A2B2C 3  6 9A9B9C18A18B18C27A···27I54A···54I81A···81AA162A···162AA
order12223699918181827···2754···5481···81162···162
size118181222222222···22···22···22···2

84 irreducible representations

dim11122222222
type+++++++++++
imageC1C2C2S3D6D9D18D27D54D81D162
kernelD162D81C162C54C27C18C9C6C3C2C1
# reps1211133992727

Matrix representation of D162 in GL3(𝔽163) generated by

16200
011016
014794
,
100
011016
06953
G:=sub<GL(3,GF(163))| [162,0,0,0,110,147,0,16,94],[1,0,0,0,110,69,0,16,53] >;

D162 in GAP, Magma, Sage, TeX

D_{162}
% in TeX

G:=Group("D162");
// GroupNames label

G:=SmallGroup(324,4);
// by ID

G=gap.SmallGroup(324,4);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,362,284,1443,381,5404,208,7781]);
// Polycyclic

G:=Group<a,b|a^162=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D162 in TeX

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