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G = D4×C40order 320 = 26·5

Direct product of C40 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C40, C41(C2×C40), (C4×C40)⋊9C2, (C4×C8)⋊4C10, C4⋊C818C10, C2012(C2×C8), C2.3(D4×C20), C4⋊C4.11C20, C221(C2×C40), (C22×C40)⋊9C2, (C22×C8)⋊5C10, C4.79(D4×C10), C22⋊C815C10, (D4×C20).29C2, (C2×D4).11C20, (D4×C10).36C4, (C4×D4).14C10, C10.141(C4×D4), C20.484(C2×D4), C22⋊C4.7C20, C2.4(C22×C40), C10.72(C8○D4), C42.68(C2×C10), C23.18(C2×C20), C10.57(C22×C8), C20.353(C4○D4), (C2×C40).361C22, (C2×C20).990C23, (C4×C20).353C22, C22.22(C22×C20), (C22×C20).499C22, (C5×C4⋊C8)⋊37C2, C2.2(C5×C8○D4), (C2×C10)⋊10(C2×C8), (C5×C4⋊C4).36C4, C4.51(C5×C4○D4), (C5×C22⋊C8)⋊32C2, (C2×C4).36(C2×C20), (C2×C20).385(C2×C4), (C2×C8).107(C2×C10), (C5×C22⋊C4).22C4, (C22×C4).95(C2×C10), (C2×C10).341(C22×C4), (C2×C4).158(C22×C10), (C22×C10).153(C2×C4), SmallGroup(320,935)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C40
C1C2C4C2×C4C2×C20C2×C40C5×C22⋊C8 — D4×C40
C1C2 — D4×C40
C1C2×C40 — D4×C40

Generators and relations for D4×C40
 G = < a,b,c | a40=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 178 in 134 conjugacy classes, 90 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×4], C22 [×4], C5, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], C22×C4 [×2], C2×D4, C20 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C40 [×2], C40 [×3], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×C10 [×2], C8×D4, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C2×C40 [×2], C2×C40 [×4], C22×C20 [×2], D4×C10, C4×C40, C5×C22⋊C8 [×2], C5×C4⋊C8, D4×C20, C22×C40 [×2], D4×C40
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], D4 [×2], C23, C10 [×7], C2×C8 [×6], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C22×C8, C8○D4, C40 [×4], C2×C20 [×6], C5×D4 [×2], C22×C10, C8×D4, C2×C40 [×6], C22×C20, D4×C10, C5×C4○D4, D4×C20, C22×C40, C5×C8○D4, D4×C40

Smallest permutation representation of D4×C40
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)
(1 148 69 87)(2 149 70 88)(3 150 71 89)(4 151 72 90)(5 152 73 91)(6 153 74 92)(7 154 75 93)(8 155 76 94)(9 156 77 95)(10 157 78 96)(11 158 79 97)(12 159 80 98)(13 160 41 99)(14 121 42 100)(15 122 43 101)(16 123 44 102)(17 124 45 103)(18 125 46 104)(19 126 47 105)(20 127 48 106)(21 128 49 107)(22 129 50 108)(23 130 51 109)(24 131 52 110)(25 132 53 111)(26 133 54 112)(27 134 55 113)(28 135 56 114)(29 136 57 115)(30 137 58 116)(31 138 59 117)(32 139 60 118)(33 140 61 119)(34 141 62 120)(35 142 63 81)(36 143 64 82)(37 144 65 83)(38 145 66 84)(39 146 67 85)(40 147 68 86)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 160)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 128)(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)(65 144)(66 145)(67 146)(68 147)(69 148)(70 149)(71 150)(72 151)(73 152)(74 153)(75 154)(76 155)(77 156)(78 157)(79 158)(80 159)

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

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

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

200 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4L5A5B5C5D8A···8H8I···8T10A···10L10M···10AB20A···20P20Q···20AV40A···40AF40AG···40CB
order1222222244444···455558···88···810···1010···1020···2020···2040···4040···40
size1111222211112···211111···12···21···12···21···12···21···12···2

200 irreducible representations

dim11111111111111111111222222
type+++++++
imageC1C2C2C2C2C2C4C4C4C5C8C10C10C10C10C10C20C20C20C40D4C4○D4C8○D4C5×D4C5×C4○D4C5×C8○D4
kernelD4×C40C4×C40C5×C22⋊C8C5×C4⋊C8D4×C20C22×C40C5×C22⋊C4C5×C4⋊C4D4×C10C8×D4C5×D4C4×C8C22⋊C8C4⋊C8C4×D4C22×C8C22⋊C4C4⋊C4C2×D4D4C40C20C10C8C4C2
# reps112112422416484481688642248816

Matrix representation of D4×C40 in GL3(𝔽41) generated by

3800
020
002
,
4000
0118
0940
,
100
0118
0040
G:=sub<GL(3,GF(41))| [38,0,0,0,2,0,0,0,2],[40,0,0,0,1,9,0,18,40],[1,0,0,0,1,0,0,18,40] >;

D4×C40 in GAP, Magma, Sage, TeX

D_4\times C_{40}
% in TeX

G:=Group("D4xC40");
// GroupNames label

G:=SmallGroup(320,935);
// by ID

G=gap.SmallGroup(320,935);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,436,124]);
// Polycyclic

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

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