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G = D5×C32order 320 = 26·5

Direct product of C32 and D5

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

Aliases: D5×C32, C1605C2, D10.4C16, C16.19D10, C80.24C22, Dic5.4C16, C53(C2×C32), C52C326C2, C52C8.9C8, (C4×D5).9C8, C4.16(C8×D5), C2.1(D5×C16), C8.36(C4×D5), C52C16.7C4, 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

C1C5 — D5×C32
C1C5C10C20C40C80D5×C16 — D5×C32
C5 — D5×C32
C1C32

Generators and relations for D5×C32
 G = < a,b,c | a32=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C22
5C4
5C2×C4
5C8
5C2×C8
5C16
5C2×C16
5C32
5C2×C32

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)
(1 154 74 59 123)(2 155 75 60 124)(3 156 76 61 125)(4 157 77 62 126)(5 158 78 63 127)(6 159 79 64 128)(7 160 80 33 97)(8 129 81 34 98)(9 130 82 35 99)(10 131 83 36 100)(11 132 84 37 101)(12 133 85 38 102)(13 134 86 39 103)(14 135 87 40 104)(15 136 88 41 105)(16 137 89 42 106)(17 138 90 43 107)(18 139 91 44 108)(19 140 92 45 109)(20 141 93 46 110)(21 142 94 47 111)(22 143 95 48 112)(23 144 96 49 113)(24 145 65 50 114)(25 146 66 51 115)(26 147 67 52 116)(27 148 68 53 117)(28 149 69 54 118)(29 150 70 55 119)(30 151 71 56 120)(31 152 72 57 121)(32 153 73 58 122)
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 113)(8 114)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 144)(34 145)(35 146)(36 147)(37 148)(38 149)(39 150)(40 151)(41 152)(42 153)(43 154)(44 155)(45 156)(46 157)(47 158)(48 159)(49 160)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(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), (1,154,74,59,123)(2,155,75,60,124)(3,156,76,61,125)(4,157,77,62,126)(5,158,78,63,127)(6,159,79,64,128)(7,160,80,33,97)(8,129,81,34,98)(9,130,82,35,99)(10,131,83,36,100)(11,132,84,37,101)(12,133,85,38,102)(13,134,86,39,103)(14,135,87,40,104)(15,136,88,41,105)(16,137,89,42,106)(17,138,90,43,107)(18,139,91,44,108)(19,140,92,45,109)(20,141,93,46,110)(21,142,94,47,111)(22,143,95,48,112)(23,144,96,49,113)(24,145,65,50,114)(25,146,66,51,115)(26,147,67,52,116)(27,148,68,53,117)(28,149,69,54,118)(29,150,70,55,119)(30,151,71,56,120)(31,152,72,57,121)(32,153,73,58,122), (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(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), (1,154,74,59,123)(2,155,75,60,124)(3,156,76,61,125)(4,157,77,62,126)(5,158,78,63,127)(6,159,79,64,128)(7,160,80,33,97)(8,129,81,34,98)(9,130,82,35,99)(10,131,83,36,100)(11,132,84,37,101)(12,133,85,38,102)(13,134,86,39,103)(14,135,87,40,104)(15,136,88,41,105)(16,137,89,42,106)(17,138,90,43,107)(18,139,91,44,108)(19,140,92,45,109)(20,141,93,46,110)(21,142,94,47,111)(22,143,95,48,112)(23,144,96,49,113)(24,145,65,50,114)(25,146,66,51,115)(26,147,67,52,116)(27,148,68,53,117)(28,149,69,54,118)(29,150,70,55,119)(30,151,71,56,120)(31,152,72,57,121)(32,153,73,58,122), (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,127)(22,128)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(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)], [(1,154,74,59,123),(2,155,75,60,124),(3,156,76,61,125),(4,157,77,62,126),(5,158,78,63,127),(6,159,79,64,128),(7,160,80,33,97),(8,129,81,34,98),(9,130,82,35,99),(10,131,83,36,100),(11,132,84,37,101),(12,133,85,38,102),(13,134,86,39,103),(14,135,87,40,104),(15,136,88,41,105),(16,137,89,42,106),(17,138,90,43,107),(18,139,91,44,108),(19,140,92,45,109),(20,141,93,46,110),(21,142,94,47,111),(22,143,95,48,112),(23,144,96,49,113),(24,145,65,50,114),(25,146,66,51,115),(26,147,67,52,116),(27,148,68,53,117),(28,149,69,54,118),(29,150,70,55,119),(30,151,71,56,120),(31,152,72,57,121),(32,153,73,58,122)], [(1,107),(2,108),(3,109),(4,110),(5,111),(6,112),(7,113),(8,114),(9,115),(10,116),(11,117),(12,118),(13,119),(14,120),(15,121),(16,122),(17,123),(18,124),(19,125),(20,126),(21,127),(22,128),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,144),(34,145),(35,146),(36,147),(37,148),(38,149),(39,150),(40,151),(41,152),(42,153),(43,154),(44,155),(45,156),(46,157),(47,158),(48,159),(49,160),(50,129),(51,130),(52,131),(53,132),(54,133),(55,134),(56,135),(57,136),(58,137),(59,138),(60,139),(61,140),(62,141),(63,142),(64,143),(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++++++
imageC1C2C2C2C4C4C8C8C16C16C32D5D10C4×D5C8×D5D5×C16D5×C32
kernelD5×C32C52C32C160D5×C16C52C16C8×D5C52C8C4×D5Dic5D10D5C32C16C8C4C2C1
# reps11112244883222481632

Matrix representation of D5×C32 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] >;

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

Subgroup lattice of D5×C32 in TeX

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