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G = D5xC32order 320 = 26·5

Direct product of C32 and D5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5xC32, C160:5C2, D10.4C16, C16.19D10, C80.24C22, Dic5.4C16, C5:3(C2xC32), C5:2C32:6C2, C5:2C8.9C8, (C4xD5).9C8, C4.16(C8xD5), C2.1(D5xC16), C8.36(C4xD5), C5:2C16.7C4, C40.94(C2xC4), C20.55(C2xC8), (C8xD5).13C4, C10.11(C2xC16), (D5xC16).11C2, SmallGroup(320,4)

Series: Derived Chief Lower central Upper central

C1C5 — D5xC32
C1C5C10C20C40C80D5xC16 — D5xC32
C5 — D5xC32
C1C32

Generators and relations for D5xC32
 G = < a,b,c | a32=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 78 in 34 conjugacy classes, 23 normal (21 characteristic)
Quotients: C1, C2, C4, C22, C8, C2xC4, D5, C16, C2xC8, D10, C32, C2xC16, C4xD5, C2xC32, C8xD5, D5xC16, D5xC32
5C2
5C2
5C22
5C4
5C2xC4
5C8
5C2xC8
5C16
5C2xC16
5C32
5C2xC32

Smallest permutation representation of D5xC32
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 2A2B2C4A4B4C4D5A5B8A8B8C8D8E8F8G8H10A10B16A···16H16I···16P20A20B20C20D32A···32P32Q···32AF40A···40H80A···80P160A···160AF
order122244445588888888101016···1616···162020202032···3232···3240···4080···80160···160
size115511552211115555221···15···522221···15···52···22···22···2

128 irreducible representations

dim11111111111222222
type++++++
imageC1C2C2C2C4C4C8C8C16C16C32D5D10C4xD5C8xD5D5xC16D5xC32
kernelD5xC32C5:2C32C160D5xC16C5:2C16C8xD5C5:2C8C4xD5Dic5D10D5C32C16C8C4C2C1
# reps11112244883222481632

Matrix representation of D5xC32 in GL2(F641) generated by

3830
0383
,
6401
361279
,
10
280640
G:=sub<GL(2,GF(641))| [383,0,0,383],[640,361,1,279],[1,280,0,640] >;

D5xC32 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

Subgroup lattice of D5xC32 in TeX

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