metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D160, C5⋊1D32, C32⋊1D5, C160⋊1C2, D80⋊1C2, 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
Generators and relations for D160
G = < a,b | a160=b2=1, bab=a-1 >
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 | 2A | 2B | 2C | 4 | 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 | 2 | 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 | 80 | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D5 | D8 | D10 | D16 | D20 | D32 | D40 | D80 | D160 |
| kernel | D160 | C160 | D80 | C40 | C32 | C20 | C16 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 32 |
Matrix representation of D160 ►in GL2(𝔽641) generated by
| 562 | 345 |
| 296 | 322 |
| 79 | 296 |
| 464 | 562 |
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
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