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## G = C160⋊C2order 320 = 26·5

### 2nd semidirect product of C160 and C2 acting faithfully

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

 Derived series C1 — C80 — C160⋊C2
 Chief series C1 — C5 — C10 — C20 — C40 — C80 — D80 — C160⋊C2
 Lower central C5 — C10 — C20 — C40 — C80 — C160⋊C2
 Upper central C1 — C2 — C4 — C8 — C16 — C32

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
5Q32
5D16
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 2A 2B 4A 4B 5A 5B 8A 8B 10A 10B 16A 16B 16C 16D 20A 20B 20C 20D 32A ··· 32H 40A ··· 40H 80A ··· 80P 160A ··· 160AF order 1 2 2 4 4 5 5 8 8 10 10 16 16 16 16 20 20 20 20 32 ··· 32 40 ··· 40 80 ··· 80 160 ··· 160 size 1 1 80 2 80 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

83 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 D4 D5 D8 D10 D16 D20 SD64 D40 D80 C160⋊C2 kernel C160⋊C2 C160 D80 Dic40 C40 C32 C20 C16 C10 C8 C5 C4 C2 C1 # reps 1 1 1 1 1 2 2 2 4 4 8 8 16 32

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

 10 5 523 5
,
 278 278 1 363
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

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