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G = D8⋊Dic5order 320 = 26·5

3rd semidirect product of D8 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D83Dic5, C4022(C2×C4), (C5×D8)⋊11C4, (C2×D8).6D5, C408C48C2, C82(C2×Dic5), C56(D8⋊C4), (D4×Dic5)⋊5C2, D42(C2×Dic5), (C10×D8).6C2, C406C420C2, (C2×C8).84D10, C10.124(C4×D4), C2.11(D4×Dic5), (C2×D4).141D10, C20.91(C4○D4), C2.7(D8⋊D5), D4⋊Dic526C2, C22.115(D4×D5), C4.27(D42D5), C4.2(C22×Dic5), C10.47(C8⋊C22), (C2×C40).146C22, (C2×C20).429C23, C20.131(C22×C4), (C2×Dic5).236D4, (D4×C10).79C22, C4⋊Dic5.163C22, (C4×Dic5).49C22, (C5×D4)⋊17(C2×C4), (C2×C10).342(C2×D4), (C2×C4).519(C22×D5), (C2×C52C8).146C22, SmallGroup(320,779)

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, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 470 in 132 conjugacy classes, 57 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C8, C2×C4, C2×C4 [×8], D4 [×4], D4 [×2], C23 [×2], C10, C10 [×2], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×8], C8⋊C4, D4⋊C4 [×2], C4.Q8, 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], D8⋊C4, C2×C52C8, C4×Dic5, C4⋊Dic5 [×2], C23.D5 [×2], C2×C40, C5×D8 [×4], C22×Dic5 [×2], D4×C10 [×2], C408C4, C406C4, 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, C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C8⋊C22 [×2], C2×Dic5 [×6], C22×D5, D8⋊C4, D4×D5, D42D5, C22×Dic5, D8⋊D5 [×2], D4×Dic5, D8⋊Dic5

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D40A···40H
order12222222444444444455888810···1010···102020202040···40
size11114444221010101020202020224420202···28···844444···4

50 irreducible representations

dim11111112222224444
type+++++++++-++-+
imageC1C2C2C2C2C2C4D4D5C4○D4D10Dic5D10C8⋊C22D42D5D4×D5D8⋊D5
kernelD8⋊Dic5C408C4C406C4D4⋊Dic5D4×Dic5C10×D8C5×D8C2×Dic5C2×D8C20C2×C8D8C2×D4C10C4C22C2
# reps11122182222842228

Matrix representation of D8⋊Dic5 in GL6(𝔽41)

4000000
0400000
00002013
00002133
00532217
0088319
,
100000
010000
00002013
00002133
00363800
00333300
,
0400000
1350000
0034100
0040000
0000401
0000535
,
13130000
9280000
0014132323
0029273133
005402629
00411215

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,5,8,0,0,0,0,3,8,0,0,20,21,22,3,0,0,13,33,17,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,33,0,0,0,0,38,33,0,0,20,21,0,0,0,0,13,33,0,0],[0,1,0,0,0,0,40,35,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,0,0,0,0,40,5,0,0,0,0,1,35],[13,9,0,0,0,0,13,28,0,0,0,0,0,0,14,29,5,4,0,0,13,27,40,1,0,0,23,31,26,12,0,0,23,33,29,15] >;

D8⋊Dic5 in GAP, Magma, Sage, TeX

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

G:=Group("D8:Dic5");
// GroupNames label

G:=SmallGroup(320,779);
// by ID

G=gap.SmallGroup(320,779);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,851,438,102,12550]);
// 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,d*a*d^-1=a^5,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽