direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C32, C160⋊5C2, D10.4C16, C16.19D10, C80.24C22, Dic5.4C16, C5⋊3(C2×C32), C5⋊2C32⋊6C2, C5⋊2C8.9C8, (C4×D5).9C8, C4.16(C8×D5), C2.1(D5×C16), C8.36(C4×D5), C5⋊2C16.7C4, C40.94(C2×C4), C20.55(C2×C8), (C8×D5).13C4, C10.11(C2×C16), (D5×C16).11C2, SmallGroup(320,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C32 |
Generators and relations for D5×C32
G = < a,b,c | a32=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C32
►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)
(1 142 73 43 118)(2 143 74 44 119)(3 144 75 45 120)(4 145 76 46 121)(5 146 77 47 122)(6 147 78 48 123)(7 148 79 49 124)(8 149 80 50 125)(9 150 81 51 126)(10 151 82 52 127)(11 152 83 53 128)(12 153 84 54 97)(13 154 85 55 98)(14 155 86 56 99)(15 156 87 57 100)(16 157 88 58 101)(17 158 89 59 102)(18 159 90 60 103)(19 160 91 61 104)(20 129 92 62 105)(21 130 93 63 106)(22 131 94 64 107)(23 132 95 33 108)(24 133 96 34 109)(25 134 65 35 110)(26 135 66 36 111)(27 136 67 37 112)(28 137 68 38 113)(29 138 69 39 114)(30 139 70 40 115)(31 140 71 41 116)(32 141 72 42 117)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 121)(21 122)(22 123)(23 124)(24 125)(25 126)(26 127)(27 128)(28 97)(29 98)(30 99)(31 100)(32 101)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(41 156)(42 157)(43 158)(44 159)(45 160)(46 129)(47 130)(48 131)(49 132)(50 133)(51 134)(52 135)(53 136)(54 137)(55 138)(56 139)(57 140)(58 141)(59 142)(60 143)(61 144)(62 145)(63 146)(64 147)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)
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), (1,142,73,43,118)(2,143,74,44,119)(3,144,75,45,120)(4,145,76,46,121)(5,146,77,47,122)(6,147,78,48,123)(7,148,79,49,124)(8,149,80,50,125)(9,150,81,51,126)(10,151,82,52,127)(11,152,83,53,128)(12,153,84,54,97)(13,154,85,55,98)(14,155,86,56,99)(15,156,87,57,100)(16,157,88,58,101)(17,158,89,59,102)(18,159,90,60,103)(19,160,91,61,104)(20,129,92,62,105)(21,130,93,63,106)(22,131,94,64,107)(23,132,95,33,108)(24,133,96,34,109)(25,134,65,35,110)(26,135,66,36,111)(27,136,67,37,112)(28,137,68,38,113)(29,138,69,39,114)(30,139,70,40,115)(31,140,71,41,116)(32,141,72,42,117), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,97)(29,98)(30,99)(31,100)(32,101)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,129)(47,130)(48,131)(49,132)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,140)(58,141)(59,142)(60,143)(61,144)(62,145)(63,146)(64,147)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)>;
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), (1,142,73,43,118)(2,143,74,44,119)(3,144,75,45,120)(4,145,76,46,121)(5,146,77,47,122)(6,147,78,48,123)(7,148,79,49,124)(8,149,80,50,125)(9,150,81,51,126)(10,151,82,52,127)(11,152,83,53,128)(12,153,84,54,97)(13,154,85,55,98)(14,155,86,56,99)(15,156,87,57,100)(16,157,88,58,101)(17,158,89,59,102)(18,159,90,60,103)(19,160,91,61,104)(20,129,92,62,105)(21,130,93,63,106)(22,131,94,64,107)(23,132,95,33,108)(24,133,96,34,109)(25,134,65,35,110)(26,135,66,36,111)(27,136,67,37,112)(28,137,68,38,113)(29,138,69,39,114)(30,139,70,40,115)(31,140,71,41,116)(32,141,72,42,117), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,97)(29,98)(30,99)(31,100)(32,101)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,129)(47,130)(48,131)(49,132)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,140)(58,141)(59,142)(60,143)(61,144)(62,145)(63,146)(64,147)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96) );
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)], [(1,142,73,43,118),(2,143,74,44,119),(3,144,75,45,120),(4,145,76,46,121),(5,146,77,47,122),(6,147,78,48,123),(7,148,79,49,124),(8,149,80,50,125),(9,150,81,51,126),(10,151,82,52,127),(11,152,83,53,128),(12,153,84,54,97),(13,154,85,55,98),(14,155,86,56,99),(15,156,87,57,100),(16,157,88,58,101),(17,158,89,59,102),(18,159,90,60,103),(19,160,91,61,104),(20,129,92,62,105),(21,130,93,63,106),(22,131,94,64,107),(23,132,95,33,108),(24,133,96,34,109),(25,134,65,35,110),(26,135,66,36,111),(27,136,67,37,112),(28,137,68,38,113),(29,138,69,39,114),(30,139,70,40,115),(31,140,71,41,116),(32,141,72,42,117)], [(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,113),(13,114),(14,115),(15,116),(16,117),(17,118),(18,119),(19,120),(20,121),(21,122),(22,123),(23,124),(24,125),(25,126),(26,127),(27,128),(28,97),(29,98),(30,99),(31,100),(32,101),(33,148),(34,149),(35,150),(36,151),(37,152),(38,153),(39,154),(40,155),(41,156),(42,157),(43,158),(44,159),(45,160),(46,129),(47,130),(48,131),(49,132),(50,133),(51,134),(52,135),(53,136),(54,137),(55,138),(56,139),(57,140),(58,141),(59,142),(60,143),(61,144),(62,145),(63,146),(64,147),(65,81),(66,82),(67,83),(68,84),(69,85),(70,86),(71,87),(72,88),(73,89),(74,90),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 16A | ··· | 16H | 16I | ··· | 16P | 20A | 20B | 20C | 20D | 32A | ··· | 32P | 32Q | ··· | 32AF | 40A | ··· | 40H | 80A | ··· | 80P | 160A | ··· | 160AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 32 | ··· | 32 | 32 | ··· | 32 | 40 | ··· | 40 | 80 | ··· | 80 | 160 | ··· | 160 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 2 | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | C32 | D5 | D10 | C4×D5 | C8×D5 | D5×C16 | D5×C32 |
| kernel | D5×C32 | C5⋊2C32 | C160 | D5×C16 | C5⋊2C16 | C8×D5 | C5⋊2C8 | C4×D5 | Dic5 | D10 | D5 | C32 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 32 | 2 | 2 | 4 | 8 | 16 | 32 |
Matrix representation of D5×C32 ►in GL2(𝔽641) generated by
| 383 | 0 |
| 0 | 383 |
| 640 | 1 |
| 361 | 279 |
| 1 | 0 |
| 280 | 640 |
G:=sub<GL(2,GF(641))| [383,0,0,383],[640,361,1,279],[1,280,0,640] >;
D5×C32 in GAP, Magma, Sage, TeX
D_5\times C_{32} % in TeX
G:=Group("D5xC32"); // GroupNames label
G:=SmallGroup(320,4);
// by ID
G=gap.SmallGroup(320,4);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,36,58,80,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^32=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export