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G = D8×Dic5order 320 = 26·5

Direct product of D8 and Dic5

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

Aliases: D8×Dic5, C56(C4×D8), (C5×D8)⋊8C4, C2.5(D5×D8), C4016(C2×C4), (C2×D8).7D5, C84(C2×Dic5), (C8×Dic5)⋊4C2, D41(C2×Dic5), (D4×Dic5)⋊3C2, (C10×D8).4C2, C405C422C2, C10.42(C2×D8), C10.123(C4×D4), (C2×C8).235D10, C2.10(D4×Dic5), (C2×D4).139D10, C20.89(C4○D4), C10.31(C4○D8), D4⋊Dic524C2, C2.5(D83D5), (C2×C40).87C22, C22.114(D4×D5), C4.26(D42D5), C4.1(C22×Dic5), C20.130(C22×C4), (C2×C20).426C23, (C2×Dic5).280D4, (D4×C10).76C22, C4⋊Dic5.161C22, (C4×Dic5).268C22, (C5×D4)⋊16(C2×C4), (C2×C10).339(C2×D4), (C2×C4).516(C22×D5), (C2×C52C8).276C22, SmallGroup(320,776)

Series: Derived Chief Lower central Upper central

C1C20 — D8×Dic5
C1C5C10C2×C10C2×C20C4×Dic5D4×Dic5 — D8×Dic5
C5C10C20 — D8×Dic5
C1C22C2×C4C2×D8

Generators and relations for D8×Dic5
 G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 470 in 134 conjugacy classes, 59 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×8], C5, C8 [×2], C8, C2×C4, C2×C4 [×8], D4 [×4], D4 [×2], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×8], C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C52C8, C40 [×2], C2×Dic5 [×2], C2×Dic5 [×6], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×C10 [×2], C4×D8, C2×C52C8, C4×Dic5, C4⋊Dic5 [×2], C23.D5 [×2], C2×C40, C5×D8 [×4], C22×Dic5 [×2], D4×C10 [×2], C8×Dic5, C405C4, D4⋊Dic5 [×2], D4×Dic5 [×2], C10×D8, D8×Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, D8 [×2], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×D8, C4○D8, C2×Dic5 [×6], C22×D5, C4×D8, D4×D5, D42D5, C22×Dic5, D5×D8, D83D5, D4×Dic5, D8×Dic5

Smallest permutation representation of D8×Dic5
On 160 points
Generators in S160
(1 90 12 74 39 61 48 55)(2 81 13 75 40 62 49 56)(3 82 14 76 31 63 50 57)(4 83 15 77 32 64 41 58)(5 84 16 78 33 65 42 59)(6 85 17 79 34 66 43 60)(7 86 18 80 35 67 44 51)(8 87 19 71 36 68 45 52)(9 88 20 72 37 69 46 53)(10 89 11 73 38 70 47 54)(21 104 156 111 149 127 140 95)(22 105 157 112 150 128 131 96)(23 106 158 113 141 129 132 97)(24 107 159 114 142 130 133 98)(25 108 160 115 143 121 134 99)(26 109 151 116 144 122 135 100)(27 110 152 117 145 123 136 91)(28 101 153 118 146 124 137 92)(29 102 154 119 147 125 138 93)(30 103 155 120 148 126 139 94)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 122)(22 123)(23 124)(24 125)(25 126)(26 127)(27 128)(28 129)(29 130)(30 121)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 88)(42 89)(43 90)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(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)
(1 116 6 111)(2 115 7 120)(3 114 8 119)(4 113 9 118)(5 112 10 117)(11 123 16 128)(12 122 17 127)(13 121 18 126)(14 130 19 125)(15 129 20 124)(21 61 26 66)(22 70 27 65)(23 69 28 64)(24 68 29 63)(25 67 30 62)(31 98 36 93)(32 97 37 92)(33 96 38 91)(34 95 39 100)(35 94 40 99)(41 106 46 101)(42 105 47 110)(43 104 48 109)(44 103 49 108)(45 102 50 107)(51 155 56 160)(52 154 57 159)(53 153 58 158)(54 152 59 157)(55 151 60 156)(71 138 76 133)(72 137 77 132)(73 136 78 131)(74 135 79 140)(75 134 80 139)(81 143 86 148)(82 142 87 147)(83 141 88 146)(84 150 89 145)(85 149 90 144)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D8E8F8G8H10A···10F10G···10N20A20B20C20D40A···40H
order12222222444444444444558888888810···1010···102020202040···40
size11114444225555101020202020222222101010102···28···844444···4

56 irreducible representations

dim1111111222222224444
type++++++++++-+-++-
imageC1C2C2C2C2C2C4D4D5D8C4○D4D10Dic5D10C4○D8D42D5D4×D5D5×D8D83D5
kernelD8×Dic5C8×Dic5C405C4D4⋊Dic5D4×Dic5C10×D8C5×D8C2×Dic5C2×D8Dic5C20C2×C8D8C2×D4C10C4C22C2C2
# reps1112218224228442244

Matrix representation of D8×Dic5 in GL4(𝔽41) generated by

1000
0100
00017
001217
,
40000
04000
00024
00120
,
354000
1000
00400
00040
,
182000
352300
00320
00032
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,12,0,0,17,17],[40,0,0,0,0,40,0,0,0,0,0,12,0,0,24,0],[35,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[18,35,0,0,20,23,0,0,0,0,32,0,0,0,0,32] >;

D8×Dic5 in GAP, Magma, Sage, TeX

D_8\times {\rm Dic}_5
% in TeX

G:=Group("D8xDic5");
// GroupNames label

G:=SmallGroup(320,776);
// by ID

G=gap.SmallGroup(320,776);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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×
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