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G = D4×C2×C20order 320 = 26·5

Direct product of C2×C20 and D4

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

Aliases: D4×C2×C20, C41(C22×C20), (C23×C20)⋊6C2, C234(C2×C20), (C23×C4)⋊3C10, (C2×C42)⋊7C10, C2010(C22×C4), C4216(C2×C10), (C4×C20)⋊57C22, C2.4(C23×C20), C24.30(C2×C10), C10.77(C23×C4), C221(C22×C20), C22.59(D4×C10), (C2×C10).335C24, (C2×C20).707C23, (C22×C20)⋊58C22, (C22×D4).13C10, C10.179(C22×D4), C22.8(C23×C10), (D4×C10).330C22, C23.28(C22×C10), (C23×C10).90C22, (C22×C10).252C23, (C2×C4×C20)⋊20C2, C2.3(D4×C2×C10), (C2×C4)⋊7(C2×C20), (C2×C4⋊C4)⋊25C10, (C10×C4⋊C4)⋊52C2, C4⋊C419(C2×C10), (C2×C20)⋊45(C2×C4), (D4×C2×C10).26C2, C2.2(C10×C4○D4), (C5×C4⋊C4)⋊76C22, (C2×C10)⋊8(C22×C4), C22⋊C417(C2×C10), (C2×C22⋊C4)⋊16C10, (C10×C22⋊C4)⋊36C2, (C22×C10)⋊20(C2×C4), (C22×C4)⋊16(C2×C10), (C2×D4).76(C2×C10), C10.221(C2×C4○D4), (C2×C10).681(C2×D4), C22.27(C5×C4○D4), (C5×C22⋊C4)⋊71C22, (C2×C4).54(C22×C10), (C2×C10).227(C4○D4), SmallGroup(320,1517)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C2×C20
C1C2C22C2×C10C2×C20C5×C22⋊C4D4×C20 — D4×C2×C20
C1C2 — D4×C2×C20
C1C22×C20 — D4×C2×C20

Generators and relations for D4×C2×C20
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 578 in 426 conjugacy classes, 274 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×6], C22, C22 [×14], C22 [×24], C5, C2×C4 [×18], C2×C4 [×22], D4 [×16], C23, C23 [×12], C23 [×8], C10 [×3], C10 [×4], C10 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×12], C24 [×2], C20 [×8], C20 [×6], C2×C10, C2×C10 [×14], C2×C10 [×24], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C2×C20 [×18], C2×C20 [×22], C5×D4 [×16], C22×C10, C22×C10 [×12], C22×C10 [×8], C2×C4×D4, C4×C20 [×4], C5×C22⋊C4 [×8], C5×C4⋊C4 [×4], C22×C20 [×3], C22×C20 [×10], C22×C20 [×8], D4×C10 [×12], C23×C10 [×2], C2×C4×C20, C10×C22⋊C4 [×2], C10×C4⋊C4, D4×C20 [×8], C23×C20 [×2], D4×C2×C10, D4×C2×C20
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], D4 [×4], C23 [×15], C10 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C20 [×8], C2×C10 [×35], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C20 [×28], C5×D4 [×4], C22×C10 [×15], C2×C4×D4, C22×C20 [×14], D4×C10 [×6], C5×C4○D4 [×2], C23×C10, D4×C20 [×4], C23×C20, D4×C2×C10, C10×C4○D4, D4×C2×C20

Smallest permutation representation of D4×C2×C20
On 160 points
Generators in S160
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 144)(22 145)(23 146)(24 147)(25 148)(26 149)(27 150)(28 151)(29 152)(30 153)(31 154)(32 155)(33 156)(34 157)(35 158)(36 159)(37 160)(38 141)(39 142)(40 143)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(81 137)(82 138)(83 139)(84 140)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)
(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 47 38 91)(2 48 39 92)(3 49 40 93)(4 50 21 94)(5 51 22 95)(6 52 23 96)(7 53 24 97)(8 54 25 98)(9 55 26 99)(10 56 27 100)(11 57 28 81)(12 58 29 82)(13 59 30 83)(14 60 31 84)(15 41 32 85)(16 42 33 86)(17 43 34 87)(18 44 35 88)(19 45 36 89)(20 46 37 90)(61 113 153 139)(62 114 154 140)(63 115 155 121)(64 116 156 122)(65 117 157 123)(66 118 158 124)(67 119 159 125)(68 120 160 126)(69 101 141 127)(70 102 142 128)(71 103 143 129)(72 104 144 130)(73 105 145 131)(74 106 146 132)(75 107 147 133)(76 108 148 134)(77 109 149 135)(78 110 150 136)(79 111 151 137)(80 112 152 138)
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(81 137)(82 138)(83 139)(84 140)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)

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

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

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

200 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4X5A5B5C5D10A···10AB10AC···10BH20A···20AF20AG···20CR
order12···22···24···44···4555510···1010···1020···2020···20
size11···12···21···12···211111···12···21···12···2

200 irreducible representations

dim11111111111111112222
type++++++++
imageC1C2C2C2C2C2C2C4C5C10C10C10C10C10C10C20D4C4○D4C5×D4C5×C4○D4
kernelD4×C2×C20C2×C4×C20C10×C22⋊C4C10×C4⋊C4D4×C20C23×C20D4×C2×C10D4×C10C2×C4×D4C2×C42C2×C22⋊C4C2×C4⋊C4C4×D4C23×C4C22×D4C2×D4C2×C20C2×C10C2×C4C22
# reps1121821164484328464441616

Matrix representation of D4×C2×C20 in GL4(𝔽41) generated by

1000
04000
0010
0001
,
20000
0900
0020
0002
,
40000
04000
00402
00401
,
40000
0100
00400
00401
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[20,0,0,0,0,9,0,0,0,0,2,0,0,0,0,2],[40,0,0,0,0,40,0,0,0,0,40,40,0,0,2,1],[40,0,0,0,0,1,0,0,0,0,40,40,0,0,0,1] >;

D4×C2×C20 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_{20}
% in TeX

G:=Group("D4xC2xC20");
// GroupNames label

G:=SmallGroup(320,1517);
// by ID

G=gap.SmallGroup(320,1517);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,856]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽