metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C32⋊2D5, C160⋊2C2, C5⋊1SD64, C8.6D20, C2.4D80, C4.2D40, D80.1C2, C40.56D4, C20.27D8, C10.2D16, Dic40⋊1C2, C16.14D10, C80.15C22, SmallGroup(320,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C160⋊C2
G = < a,b | a160=b2=1, bab=a79 >
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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