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G = C160⋊C2order 320 = 26·5

2nd semidirect product of C160 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C322D5, C1602C2, C51SD64, C8.6D20, C2.4D80, C4.2D40, D80.1C2, C40.56D4, C20.27D8, C10.2D16, Dic401C2, C16.14D10, C80.15C22, SmallGroup(320,7)

Series: Derived Chief Lower central Upper central

C1C80 — C160⋊C2
C1C5C10C20C40C80D80 — C160⋊C2
C5C10C20C40C80 — C160⋊C2
C1C2C4C8C16C32

Generators and relations for C160⋊C2
 G = < a,b | a160=b2=1, bab=a79 >

80C2
40C22
40C4
16D5
20Q8
20D4
8D10
8Dic5
10Q16
10D8
4D20
4Dic10
5Q32
5D16
2Dic20
2D40
5SD64

Smallest permutation representation of C160⋊C2
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)
(2 80)(3 159)(4 78)(5 157)(6 76)(7 155)(8 74)(9 153)(10 72)(11 151)(12 70)(13 149)(14 68)(15 147)(16 66)(17 145)(18 64)(19 143)(20 62)(21 141)(22 60)(23 139)(24 58)(25 137)(26 56)(27 135)(28 54)(29 133)(30 52)(31 131)(32 50)(33 129)(34 48)(35 127)(36 46)(37 125)(38 44)(39 123)(40 42)(41 121)(43 119)(45 117)(47 115)(49 113)(51 111)(53 109)(55 107)(57 105)(59 103)(61 101)(63 99)(65 97)(67 95)(69 93)(71 91)(73 89)(75 87)(77 85)(79 83)(82 160)(84 158)(86 156)(88 154)(90 152)(92 150)(94 148)(96 146)(98 144)(100 142)(102 140)(104 138)(106 136)(108 134)(110 132)(112 130)(114 128)(116 126)(118 124)(120 122)

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

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

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

83 conjugacy classes

class 1 2A2B4A4B5A5B8A8B10A10B16A16B16C16D20A20B20C20D32A···32H40A···40H80A···80P160A···160AF
order1224455881010161616162020202032···3240···4080···80160···160
size1180280222222222222222···22···22···22···2

83 irreducible representations

dim11112222222222
type++++++++++++
imageC1C2C2C2D4D5D8D10D16D20SD64D40D80C160⋊C2
kernelC160⋊C2C160D80Dic40C40C32C20C16C10C8C5C4C2C1
# reps1111122244881632

Matrix representation of C160⋊C2 in GL2(𝔽641) generated by

105
5235
,
278278
1363
G:=sub<GL(2,GF(641))| [10,523,5,5],[278,1,278,363] >;

C160⋊C2 in GAP, Magma, Sage, TeX

C_{160}\rtimes C_2
% in TeX

G:=Group("C160:C2");
// GroupNames label

G:=SmallGroup(320,7);
// by ID

G=gap.SmallGroup(320,7);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,926,142,1571,80,1684,102,12550]);
// Polycyclic

G:=Group<a,b|a^160=b^2=1,b*a*b=a^79>;
// generators/relations

Export

Subgroup lattice of C160⋊C2 in TeX

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