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G = D160order 320 = 26·5

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D160, C51D32, C321D5, C1601C2, D801C2, C8.5D20, C4.1D40, C2.3D80, C40.55D4, C10.1D16, C20.26D8, C16.13D10, C80.14C22, sometimes denoted D320 or Dih160 or Dih320, SmallGroup(320,6)

Series: Derived Chief Lower central Upper central

C1C80 — D160
C1C5C10C20C40C80D80 — D160
C5C10C20C40C80 — D160
C1C2C4C8C16C32

Generators and relations for D160
 G = < a,b | a160=b2=1, bab=a-1 >

80C2
80C2
40C22
40C22
16D5
16D5
20D4
20D4
8D10
8D10
10D8
10D8
4D20
4D20
5D16
5D16
2D40
2D40
5D32

Smallest permutation representation of D160
On 160 points
Generators in S160
(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)
(1 160)(2 159)(3 158)(4 157)(5 156)(6 155)(7 154)(8 153)(9 152)(10 151)(11 150)(12 149)(13 148)(14 147)(15 146)(16 145)(17 144)(18 143)(19 142)(20 141)(21 140)(22 139)(23 138)(24 137)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 128)(34 127)(35 126)(36 125)(37 124)(38 123)(39 122)(40 121)(41 120)(42 119)(43 118)(44 117)(45 116)(46 115)(47 114)(48 113)(49 112)(50 111)(51 110)(52 109)(53 108)(54 107)(55 106)(56 105)(57 104)(58 103)(59 102)(60 101)(61 100)(62 99)(63 98)(64 97)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)

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

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

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

83 conjugacy classes

class 1 2A2B2C 4 5A5B8A8B10A10B16A16B16C16D20A20B20C20D32A···32H40A···40H80A···80P160A···160AF
order1222455881010161616162020202032···3240···4080···80160···160
size1180802222222222222222···22···22···22···2

83 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2D4D5D8D10D16D20D32D40D80D160
kernelD160C160D80C40C32C20C16C10C8C5C4C2C1
# reps112122244881632

Matrix representation of D160 in GL2(𝔽641) generated by

562345
296322
,
79296
464562
G:=sub<GL(2,GF(641))| [562,296,345,322],[79,464,296,562] >;

D160 in GAP, Magma, Sage, TeX

D_{160}
% in TeX

G:=Group("D160");
// GroupNames label

G:=SmallGroup(320,6);
// by ID

G=gap.SmallGroup(320,6);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,254,142,675,192,1684,102,12550]);
// Polycyclic

G:=Group<a,b|a^160=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D160 in TeX

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