metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C32⋊3D5, C160⋊6C2, C5⋊3M6(2), D10.1C16, C16.20D10, C80.25C22, Dic5.1C16, C5⋊2C32⋊4C2, C5⋊2C8.2C8, (C4×D5).2C8, (C8×D5).2C4, C8.37(C4×D5), C4.17(C8×D5), C2.3(D5×C16), C5⋊2C16.2C4, C20.56(C2×C8), C40.95(C2×C4), (D5×C16).4C2, C10.12(C2×C16), SmallGroup(320,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊D5
G = < a,b,c | a32=b5=c2=1, ab=ba, cac=a17, cbc=b-1 >
Smallest permutation representation of C32⋊D5
►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)
(1 104 80 62 143)(2 105 81 63 144)(3 106 82 64 145)(4 107 83 33 146)(5 108 84 34 147)(6 109 85 35 148)(7 110 86 36 149)(8 111 87 37 150)(9 112 88 38 151)(10 113 89 39 152)(11 114 90 40 153)(12 115 91 41 154)(13 116 92 42 155)(14 117 93 43 156)(15 118 94 44 157)(16 119 95 45 158)(17 120 96 46 159)(18 121 65 47 160)(19 122 66 48 129)(20 123 67 49 130)(21 124 68 50 131)(22 125 69 51 132)(23 126 70 52 133)(24 127 71 53 134)(25 128 72 54 135)(26 97 73 55 136)(27 98 74 56 137)(28 99 75 57 138)(29 100 76 58 139)(30 101 77 59 140)(31 102 78 60 141)(32 103 79 61 142)
(1 143)(2 160)(3 145)(4 130)(5 147)(6 132)(7 149)(8 134)(9 151)(10 136)(11 153)(12 138)(13 155)(14 140)(15 157)(16 142)(17 159)(18 144)(19 129)(20 146)(21 131)(22 148)(23 133)(24 150)(25 135)(26 152)(27 137)(28 154)(29 139)(30 156)(31 141)(32 158)(33 123)(34 108)(35 125)(36 110)(37 127)(38 112)(39 97)(40 114)(41 99)(42 116)(43 101)(44 118)(45 103)(46 120)(47 105)(48 122)(49 107)(50 124)(51 109)(52 126)(53 111)(54 128)(55 113)(56 98)(57 115)(58 100)(59 117)(60 102)(61 119)(62 104)(63 121)(64 106)(65 81)(67 83)(69 85)(71 87)(73 89)(75 91)(77 93)(79 95)
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), (1,104,80,62,143)(2,105,81,63,144)(3,106,82,64,145)(4,107,83,33,146)(5,108,84,34,147)(6,109,85,35,148)(7,110,86,36,149)(8,111,87,37,150)(9,112,88,38,151)(10,113,89,39,152)(11,114,90,40,153)(12,115,91,41,154)(13,116,92,42,155)(14,117,93,43,156)(15,118,94,44,157)(16,119,95,45,158)(17,120,96,46,159)(18,121,65,47,160)(19,122,66,48,129)(20,123,67,49,130)(21,124,68,50,131)(22,125,69,51,132)(23,126,70,52,133)(24,127,71,53,134)(25,128,72,54,135)(26,97,73,55,136)(27,98,74,56,137)(28,99,75,57,138)(29,100,76,58,139)(30,101,77,59,140)(31,102,78,60,141)(32,103,79,61,142), (1,143)(2,160)(3,145)(4,130)(5,147)(6,132)(7,149)(8,134)(9,151)(10,136)(11,153)(12,138)(13,155)(14,140)(15,157)(16,142)(17,159)(18,144)(19,129)(20,146)(21,131)(22,148)(23,133)(24,150)(25,135)(26,152)(27,137)(28,154)(29,139)(30,156)(31,141)(32,158)(33,123)(34,108)(35,125)(36,110)(37,127)(38,112)(39,97)(40,114)(41,99)(42,116)(43,101)(44,118)(45,103)(46,120)(47,105)(48,122)(49,107)(50,124)(51,109)(52,126)(53,111)(54,128)(55,113)(56,98)(57,115)(58,100)(59,117)(60,102)(61,119)(62,104)(63,121)(64,106)(65,81)(67,83)(69,85)(71,87)(73,89)(75,91)(77,93)(79,95)>;
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), (1,104,80,62,143)(2,105,81,63,144)(3,106,82,64,145)(4,107,83,33,146)(5,108,84,34,147)(6,109,85,35,148)(7,110,86,36,149)(8,111,87,37,150)(9,112,88,38,151)(10,113,89,39,152)(11,114,90,40,153)(12,115,91,41,154)(13,116,92,42,155)(14,117,93,43,156)(15,118,94,44,157)(16,119,95,45,158)(17,120,96,46,159)(18,121,65,47,160)(19,122,66,48,129)(20,123,67,49,130)(21,124,68,50,131)(22,125,69,51,132)(23,126,70,52,133)(24,127,71,53,134)(25,128,72,54,135)(26,97,73,55,136)(27,98,74,56,137)(28,99,75,57,138)(29,100,76,58,139)(30,101,77,59,140)(31,102,78,60,141)(32,103,79,61,142), (1,143)(2,160)(3,145)(4,130)(5,147)(6,132)(7,149)(8,134)(9,151)(10,136)(11,153)(12,138)(13,155)(14,140)(15,157)(16,142)(17,159)(18,144)(19,129)(20,146)(21,131)(22,148)(23,133)(24,150)(25,135)(26,152)(27,137)(28,154)(29,139)(30,156)(31,141)(32,158)(33,123)(34,108)(35,125)(36,110)(37,127)(38,112)(39,97)(40,114)(41,99)(42,116)(43,101)(44,118)(45,103)(46,120)(47,105)(48,122)(49,107)(50,124)(51,109)(52,126)(53,111)(54,128)(55,113)(56,98)(57,115)(58,100)(59,117)(60,102)(61,119)(62,104)(63,121)(64,106)(65,81)(67,83)(69,85)(71,87)(73,89)(75,91)(77,93)(79,95) );
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)], [(1,104,80,62,143),(2,105,81,63,144),(3,106,82,64,145),(4,107,83,33,146),(5,108,84,34,147),(6,109,85,35,148),(7,110,86,36,149),(8,111,87,37,150),(9,112,88,38,151),(10,113,89,39,152),(11,114,90,40,153),(12,115,91,41,154),(13,116,92,42,155),(14,117,93,43,156),(15,118,94,44,157),(16,119,95,45,158),(17,120,96,46,159),(18,121,65,47,160),(19,122,66,48,129),(20,123,67,49,130),(21,124,68,50,131),(22,125,69,51,132),(23,126,70,52,133),(24,127,71,53,134),(25,128,72,54,135),(26,97,73,55,136),(27,98,74,56,137),(28,99,75,57,138),(29,100,76,58,139),(30,101,77,59,140),(31,102,78,60,141),(32,103,79,61,142)], [(1,143),(2,160),(3,145),(4,130),(5,147),(6,132),(7,149),(8,134),(9,151),(10,136),(11,153),(12,138),(13,155),(14,140),(15,157),(16,142),(17,159),(18,144),(19,129),(20,146),(21,131),(22,148),(23,133),(24,150),(25,135),(26,152),(27,137),(28,154),(29,139),(30,156),(31,141),(32,158),(33,123),(34,108),(35,125),(36,110),(37,127),(38,112),(39,97),(40,114),(41,99),(42,116),(43,101),(44,118),(45,103),(46,120),(47,105),(48,122),(49,107),(50,124),(51,109),(52,126),(53,111),(54,128),(55,113),(56,98),(57,115),(58,100),(59,117),(60,102),(61,119),(62,104),(63,121),(64,106),(65,81),(67,83),(69,85),(71,87),(73,89),(75,91),(77,93),(79,95)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 20A | 20B | 20C | 20D | 32A | ··· | 32H | 32I | ··· | 32P | 40A | ··· | 40H | 80A | ··· | 80P | 160A | ··· | 160AF |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 32 | ··· | 32 | 32 | ··· | 32 | 40 | ··· | 40 | 80 | ··· | 80 | 160 | ··· | 160 |
| size | 1 | 1 | 10 | 1 | 1 | 10 | 2 | 2 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | D5 | D10 | C4×D5 | M6(2) | C8×D5 | D5×C16 | C32⋊D5 |
| kernel | C32⋊D5 | C5⋊2C32 | C160 | D5×C16 | C5⋊2C16 | C8×D5 | C5⋊2C8 | C4×D5 | Dic5 | D10 | C32 | C16 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 8 | 8 | 16 | 32 |
Matrix representation of C32⋊D5 ►in GL2(𝔽641) generated by
| 540 | 252 |
| 389 | 101 |
| 0 | 1 |
| 640 | 362 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(641))| [540,389,252,101],[0,640,1,362],[0,1,1,0] >;
C32⋊D5 in GAP, Magma, Sage, TeX
C_{32}\rtimes D_5 % in TeX
G:=Group("C32:D5"); // GroupNames label
G:=SmallGroup(320,5);
// by ID
G=gap.SmallGroup(320,5);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,36,58,80,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^32=b^5=c^2=1,a*b=b*a,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations
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