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G = C4×D40order 320 = 26·5

Direct product of C4 and D40

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×D40, C205D8, C42.259D10, C52(C4×D8), (C4×C8)⋊7D5, C810(C4×D5), C4032(C2×C4), (C4×C40)⋊12C2, (C4×D20)⋊1C2, C10.2(C2×D8), C2.1(C2×D40), D2018(C2×C4), C405C428C2, C2.10(C4×D20), C10.37(C4×D4), (C2×C4).61D20, (C2×D40).14C2, C10.3(C4○D8), (C2×C8).286D10, (C2×C20).351D4, D205C443C2, C22.28(C2×D20), C20.217(C4○D4), C4.101(C4○D20), C2.2(D407C2), (C4×C20).326C22, (C2×C40).346C22, (C2×C20).721C23, C20.161(C22×C4), (C2×D20).194C22, C4⋊Dic5.263C22, C4.60(C2×C4×D5), (C2×C10).104(C2×D4), (C2×C4).664(C22×D5), SmallGroup(320,319)

Series: Derived Chief Lower central Upper central

C1C20 — C4×D40
C1C5C10C2×C10C2×C20C2×D20C2×D40 — C4×D40
C5C10C20 — C4×D40
C1C2×C4C42C4×C8

Generators and relations for C4×D40
 G = < a,b,c | a4=b40=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 662 in 134 conjugacy classes, 55 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], C5, C8 [×2], C8, C2×C4 [×3], C2×C4 [×6], D4 [×6], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C20, D10 [×8], C2×C10, C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C40 [×2], C40, C4×D5 [×4], D20 [×4], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C4×D8, D40 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20 [×2], C405C4, D205C4 [×2], C4×C40, C4×D20 [×2], C2×D40, C4×D40
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, D8 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×D8, C4○D8, C4×D5 [×2], D20 [×2], C22×D5, C4×D8, D40 [×2], C2×C4×D5, C2×D20, C4○D20, C4×D20, C2×D40, D407C2, C4×D40

Smallest permutation representation of C4×D40
On 160 points
Generators in S160
(1 58 97 155)(2 59 98 156)(3 60 99 157)(4 61 100 158)(5 62 101 159)(6 63 102 160)(7 64 103 121)(8 65 104 122)(9 66 105 123)(10 67 106 124)(11 68 107 125)(12 69 108 126)(13 70 109 127)(14 71 110 128)(15 72 111 129)(16 73 112 130)(17 74 113 131)(18 75 114 132)(19 76 115 133)(20 77 116 134)(21 78 117 135)(22 79 118 136)(23 80 119 137)(24 41 120 138)(25 42 81 139)(26 43 82 140)(27 44 83 141)(28 45 84 142)(29 46 85 143)(30 47 86 144)(31 48 87 145)(32 49 88 146)(33 50 89 147)(34 51 90 148)(35 52 91 149)(36 53 92 150)(37 54 93 151)(38 55 94 152)(39 56 95 153)(40 57 96 154)
(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)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 100)(13 99)(14 98)(15 97)(16 96)(17 95)(18 94)(19 93)(20 92)(21 91)(22 90)(23 89)(24 88)(25 87)(26 86)(27 85)(28 84)(29 83)(30 82)(31 81)(32 120)(33 119)(34 118)(35 117)(36 116)(37 115)(38 114)(39 113)(40 112)(41 146)(42 145)(43 144)(44 143)(45 142)(46 141)(47 140)(48 139)(49 138)(50 137)(51 136)(52 135)(53 134)(54 133)(55 132)(56 131)(57 130)(58 129)(59 128)(60 127)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 160)(68 159)(69 158)(70 157)(71 156)(72 155)(73 154)(74 153)(75 152)(76 151)(77 150)(78 149)(79 148)(80 147)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H10A···10F20A···20X40A···40AF
order12222222444444444444558···810···1020···2040···40
size1111202020201111222220202020222···22···22···22···2

92 irreducible representations

dim1111111222222222222
type+++++++++++++
imageC1C2C2C2C2C2C4D4D5D8C4○D4D10D10C4○D8C4×D5D20D40C4○D20D407C2
kernelC4×D40C405C4D205C4C4×C40C4×D20C2×D40D40C2×C20C4×C8C20C20C42C2×C8C10C8C2×C4C4C4C2
# reps112121822422448816816

Matrix representation of C4×D40 in GL4(𝔽41) generated by

9000
0900
0010
0001
,
403500
63500
00183
003836
,
6100
63500
002516
00216
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[40,6,0,0,35,35,0,0,0,0,18,38,0,0,3,36],[6,6,0,0,1,35,0,0,0,0,25,2,0,0,16,16] >;

C4×D40 in GAP, Magma, Sage, TeX

C_4\times D_{40}
% in TeX

G:=Group("C4xD40");
// GroupNames label

G:=SmallGroup(320,319);
// by ID

G=gap.SmallGroup(320,319);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,58,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^4=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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