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G = D8×C20order 320 = 26·5

Direct product of C20 and D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: D8×C20, (C4×C8)⋊7C10, C84(C2×C20), C4036(C2×C4), (C4×C40)⋊23C2, (C4×D4)⋊1C10, D41(C2×C20), C2.3(C10×D8), (D4×C20)⋊30C2, (C2×D8).7C10, C2.D814C10, C10.75(C2×D8), C2.12(D4×C20), (C10×D8).14C2, C10.144(C4×D4), (C2×C20).360D4, C4.9(C22×C20), D4⋊C421C10, C42.70(C2×C10), C22.51(D4×C10), C20.256(C4○D4), C10.116(C4○D8), (C4×C20).355C22, C20.213(C22×C4), (C2×C40).421C22, (C2×C20).904C23, (D4×C10).290C22, C2.3(C5×C4○D8), C4.1(C5×C4○D4), (C5×D4)⋊24(C2×C4), (C5×C2.D8)⋊29C2, (C2×C4).50(C5×D4), C4⋊C4.45(C2×C10), (C2×C8).65(C2×C10), (C5×D4⋊C4)⋊44C2, (C2×D4).48(C2×C10), (C2×C10).627(C2×D4), (C5×C4⋊C4).366C22, (C2×C4).79(C22×C10), SmallGroup(320,938)

Series: Derived Chief Lower central Upper central

C1C4 — D8×C20
C1C2C22C2×C4C2×C20C5×C4⋊C4C5×D4⋊C4 — D8×C20
C1C2C4 — D8×C20
C1C2×C20C4×C20 — D8×C20

Generators and relations for D8×C20
 G = < a,b,c | a20=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 250 in 134 conjugacy classes, 74 normal (34 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 [×4], D4 [×2], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], C20 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×8], C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C40 [×2], C40, C2×C20 [×3], C2×C20 [×6], C5×D4 [×4], C5×D4 [×2], C22×C10 [×2], C4×D8, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40 [×2], C5×D8 [×4], C22×C20 [×2], D4×C10 [×2], C4×C40, C5×D4⋊C4 [×2], C5×C2.D8, D4×C20 [×2], C10×D8, D8×C20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], D8 [×2], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C2×D8, C4○D8, C2×C20 [×6], C5×D4 [×2], C22×C10, C4×D8, C5×D8 [×2], C22×C20, D4×C10, C5×C4○D4, D4×C20, C10×D8, C5×C4○D8, D8×C20

Smallest permutation representation of D8×C20
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 82 44 155 25 73 102 121)(2 83 45 156 26 74 103 122)(3 84 46 157 27 75 104 123)(4 85 47 158 28 76 105 124)(5 86 48 159 29 77 106 125)(6 87 49 160 30 78 107 126)(7 88 50 141 31 79 108 127)(8 89 51 142 32 80 109 128)(9 90 52 143 33 61 110 129)(10 91 53 144 34 62 111 130)(11 92 54 145 35 63 112 131)(12 93 55 146 36 64 113 132)(13 94 56 147 37 65 114 133)(14 95 57 148 38 66 115 134)(15 96 58 149 39 67 116 135)(16 97 59 150 40 68 117 136)(17 98 60 151 21 69 118 137)(18 99 41 152 22 70 119 138)(19 100 42 153 23 71 120 139)(20 81 43 154 24 72 101 140)
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 151)(22 152)(23 153)(24 154)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)

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

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

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

140 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B5C5D8A···8H10A···10L10M···10AB20A···20P20Q···20AF20AG···20AV40A···40AF
order1222222244444444444455558···810···1010···1020···2020···2020···2040···40
size1111444411112222444411112···21···14···41···12···24···42···2

140 irreducible representations

dim1111111111111122222222
type++++++++
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C20D4D8C4○D4C4○D8C5×D4C5×D8C5×C4○D4C5×C4○D8
kernelD8×C20C4×C40C5×D4⋊C4C5×C2.D8D4×C20C10×D8C5×D8C4×D8C4×C8D4⋊C4C2.D8C4×D4C2×D8D8C2×C20C20C20C10C2×C4C4C4C2
# reps1121218448484322424816816

Matrix representation of D8×C20 in GL3(𝔽41) generated by

3200
040
004
,
4000
02912
02929
,
100
02912
01212
G:=sub<GL(3,GF(41))| [32,0,0,0,4,0,0,0,4],[40,0,0,0,29,29,0,12,29],[1,0,0,0,29,12,0,12,12] >;

D8×C20 in GAP, Magma, Sage, TeX

D_8\times C_{20}
% in TeX

G:=Group("D8xC20");
// GroupNames label

G:=SmallGroup(320,938);
// by ID

G=gap.SmallGroup(320,938);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,436,7004,3511,172]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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