metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5⋊C32, C80.4C4, C16.5F5, D10.2C16, Dic5.2C16, C5⋊C32⋊3C2, C5⋊1(C2×C32), C5⋊2C8.5C8, (C4×D5).5C8, (C8×D5).7C4, C8.35(C2×F5), C10.1(C2×C16), C40.30(C2×C4), C20.16(C2×C8), (D5×C16).7C2, C2.1(D5⋊C16), C4.11(D5⋊C8), C5⋊2C16.13C22, SmallGroup(320,179)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5⋊C32 |
Generators and relations for D5⋊C32
G = < a,b,c | a5=b2=c32=1, bab=a-1, cac-1=a3, cbc-1=a2b >
Smallest permutation representation of D5⋊C32
►On 160 points
Generators in S160
(1 55 112 72 151)(2 73 56 152 113)(3 153 74 114 57)(4 115 154 58 75)(5 59 116 76 155)(6 77 60 156 117)(7 157 78 118 61)(8 119 158 62 79)(9 63 120 80 159)(10 81 64 160 121)(11 129 82 122 33)(12 123 130 34 83)(13 35 124 84 131)(14 85 36 132 125)(15 133 86 126 37)(16 127 134 38 87)(17 39 128 88 135)(18 89 40 136 97)(19 137 90 98 41)(20 99 138 42 91)(21 43 100 92 139)(22 93 44 140 101)(23 141 94 102 45)(24 103 142 46 95)(25 47 104 96 143)(26 65 48 144 105)(27 145 66 106 49)(28 107 146 50 67)(29 51 108 68 147)(30 69 52 148 109)(31 149 70 110 53)(32 111 150 54 71)
(1 151)(2 113)(3 57)(4 75)(5 155)(6 117)(7 61)(8 79)(9 159)(10 121)(11 33)(12 83)(13 131)(14 125)(15 37)(16 87)(17 135)(18 97)(19 41)(20 91)(21 139)(22 101)(23 45)(24 95)(25 143)(26 105)(27 49)(28 67)(29 147)(30 109)(31 53)(32 71)(34 123)(35 84)(38 127)(39 88)(42 99)(43 92)(46 103)(47 96)(50 107)(51 68)(54 111)(55 72)(58 115)(59 76)(62 119)(63 80)(65 144)(69 148)(73 152)(77 156)(81 160)(85 132)(89 136)(93 140)(98 137)(102 141)(106 145)(110 149)(114 153)(118 157)(122 129)(126 133)
(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)
G:=sub<Sym(160)| (1,55,112,72,151)(2,73,56,152,113)(3,153,74,114,57)(4,115,154,58,75)(5,59,116,76,155)(6,77,60,156,117)(7,157,78,118,61)(8,119,158,62,79)(9,63,120,80,159)(10,81,64,160,121)(11,129,82,122,33)(12,123,130,34,83)(13,35,124,84,131)(14,85,36,132,125)(15,133,86,126,37)(16,127,134,38,87)(17,39,128,88,135)(18,89,40,136,97)(19,137,90,98,41)(20,99,138,42,91)(21,43,100,92,139)(22,93,44,140,101)(23,141,94,102,45)(24,103,142,46,95)(25,47,104,96,143)(26,65,48,144,105)(27,145,66,106,49)(28,107,146,50,67)(29,51,108,68,147)(30,69,52,148,109)(31,149,70,110,53)(32,111,150,54,71), (1,151)(2,113)(3,57)(4,75)(5,155)(6,117)(7,61)(8,79)(9,159)(10,121)(11,33)(12,83)(13,131)(14,125)(15,37)(16,87)(17,135)(18,97)(19,41)(20,91)(21,139)(22,101)(23,45)(24,95)(25,143)(26,105)(27,49)(28,67)(29,147)(30,109)(31,53)(32,71)(34,123)(35,84)(38,127)(39,88)(42,99)(43,92)(46,103)(47,96)(50,107)(51,68)(54,111)(55,72)(58,115)(59,76)(62,119)(63,80)(65,144)(69,148)(73,152)(77,156)(81,160)(85,132)(89,136)(93,140)(98,137)(102,141)(106,145)(110,149)(114,153)(118,157)(122,129)(126,133), (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)>;
G:=Group( (1,55,112,72,151)(2,73,56,152,113)(3,153,74,114,57)(4,115,154,58,75)(5,59,116,76,155)(6,77,60,156,117)(7,157,78,118,61)(8,119,158,62,79)(9,63,120,80,159)(10,81,64,160,121)(11,129,82,122,33)(12,123,130,34,83)(13,35,124,84,131)(14,85,36,132,125)(15,133,86,126,37)(16,127,134,38,87)(17,39,128,88,135)(18,89,40,136,97)(19,137,90,98,41)(20,99,138,42,91)(21,43,100,92,139)(22,93,44,140,101)(23,141,94,102,45)(24,103,142,46,95)(25,47,104,96,143)(26,65,48,144,105)(27,145,66,106,49)(28,107,146,50,67)(29,51,108,68,147)(30,69,52,148,109)(31,149,70,110,53)(32,111,150,54,71), (1,151)(2,113)(3,57)(4,75)(5,155)(6,117)(7,61)(8,79)(9,159)(10,121)(11,33)(12,83)(13,131)(14,125)(15,37)(16,87)(17,135)(18,97)(19,41)(20,91)(21,139)(22,101)(23,45)(24,95)(25,143)(26,105)(27,49)(28,67)(29,147)(30,109)(31,53)(32,71)(34,123)(35,84)(38,127)(39,88)(42,99)(43,92)(46,103)(47,96)(50,107)(51,68)(54,111)(55,72)(58,115)(59,76)(62,119)(63,80)(65,144)(69,148)(73,152)(77,156)(81,160)(85,132)(89,136)(93,140)(98,137)(102,141)(106,145)(110,149)(114,153)(118,157)(122,129)(126,133), (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) );
G=PermutationGroup([[(1,55,112,72,151),(2,73,56,152,113),(3,153,74,114,57),(4,115,154,58,75),(5,59,116,76,155),(6,77,60,156,117),(7,157,78,118,61),(8,119,158,62,79),(9,63,120,80,159),(10,81,64,160,121),(11,129,82,122,33),(12,123,130,34,83),(13,35,124,84,131),(14,85,36,132,125),(15,133,86,126,37),(16,127,134,38,87),(17,39,128,88,135),(18,89,40,136,97),(19,137,90,98,41),(20,99,138,42,91),(21,43,100,92,139),(22,93,44,140,101),(23,141,94,102,45),(24,103,142,46,95),(25,47,104,96,143),(26,65,48,144,105),(27,145,66,106,49),(28,107,146,50,67),(29,51,108,68,147),(30,69,52,148,109),(31,149,70,110,53),(32,111,150,54,71)], [(1,151),(2,113),(3,57),(4,75),(5,155),(6,117),(7,61),(8,79),(9,159),(10,121),(11,33),(12,83),(13,131),(14,125),(15,37),(16,87),(17,135),(18,97),(19,41),(20,91),(21,139),(22,101),(23,45),(24,95),(25,143),(26,105),(27,49),(28,67),(29,147),(30,109),(31,53),(32,71),(34,123),(35,84),(38,127),(39,88),(42,99),(43,92),(46,103),(47,96),(50,107),(51,68),(54,111),(55,72),(58,115),(59,76),(62,119),(63,80),(65,144),(69,148),(73,152),(77,156),(81,160),(85,132),(89,136),(93,140),(98,137),(102,141),(106,145),(110,149),(114,153),(118,157),(122,129),(126,133)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 16A | ··· | 16H | 16I | ··· | 16P | 20A | 20B | 32A | ··· | 32AF | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | 20 | 32 | ··· | 32 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 4 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | C32 | F5 | C2×F5 | D5⋊C8 | D5⋊C16 | D5⋊C32 |
| kernel | D5⋊C32 | C5⋊C32 | D5×C16 | C80 | C8×D5 | C5⋊2C8 | C4×D5 | Dic5 | D10 | D5 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 32 | 1 | 1 | 2 | 4 | 8 |
Matrix representation of D5⋊C32 ►in GL4(𝔽641) generated by
| 640 | 1 | 0 | 0 |
| 640 | 0 | 1 | 0 |
| 640 | 0 | 0 | 1 |
| 640 | 0 | 0 | 0 |
| 0 | 1 | 640 | 0 |
| 1 | 0 | 640 | 0 |
| 0 | 0 | 640 | 0 |
| 0 | 0 | 640 | 1 |
| 619 | 0 | 22 | 221 |
| 0 | 221 | 442 | 199 |
| 420 | 199 | 442 | 221 |
| 420 | 221 | 22 | 0 |
G:=sub<GL(4,GF(641))| [640,640,640,640,1,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,1,0,0,0,640,640,640,640,0,0,0,1],[619,0,420,420,0,221,199,221,22,442,442,22,221,199,221,0] >;
D5⋊C32 in GAP, Magma, Sage, TeX
D_5\rtimes C_{32} % in TeX
G:=Group("D5:C32"); // GroupNames label
G:=SmallGroup(320,179);
// by ID
G=gap.SmallGroup(320,179);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,64,58,80,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^5=b^2=c^32=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^2*b>;
// generators/relations
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