Copied to
clipboard

G = Dic54D8order 320 = 26·5

1st semidirect product of Dic5 and D8 acting through Inn(Dic5)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic54D8, C53(C4×D8), D4⋊D55C4, D41(C4×D5), C2.1(D5×D8), D2011(C2×C4), (D4×Dic5)⋊1C2, C10.60(C4×D4), C10.18(C2×D8), D208C41C2, D4⋊C421D5, C4⋊C4.128D10, (C8×Dic5)⋊17C2, (C2×C8).197D10, C10.D81C2, D205C414C2, C22.65(D4×D5), (C2×D4).123D10, C10.36(C4○D8), C20.36(C22×C4), C20.142(C4○D4), C4.43(D42D5), (C2×C40).175C22, (C2×C20).196C23, (C2×Dic5).267D4, (D4×C10).17C22, (C2×D20).47C22, C4⋊Dic5.56C22, C2.1(SD163D5), C2.14(Dic54D4), (C4×Dic5).251C22, C4.1(C2×C4×D5), C52C817(C2×C4), (C5×D4)⋊11(C2×C4), (C2×D4⋊D5).1C2, (C5×C4⋊C4).1C22, (C5×D4⋊C4)⋊15C2, (C2×C10).209(C2×D4), (C2×C4).303(C22×D5), (C2×C52C8).215C22, SmallGroup(320,383)

Series: Derived Chief Lower central Upper central

C1C20 — Dic54D8
C1C5C10C20C2×C20C4×Dic5D4×Dic5 — Dic54D8
C5C10C20 — Dic54D8
C1C22C2×C4D4⋊C4

Generators and relations for Dic54D8
 G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 566 in 134 conjugacy classes, 51 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×8], C5, C8 [×3], C2×C4, C2×C4 [×8], D4 [×2], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×4], C22×C4 [×2], C2×D4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×4], C2×C10, C2×C10 [×4], C4×C8, D4⋊C4, D4⋊C4, C2.D8, C4×D4 [×2], C2×D8, C52C8 [×2], C40, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5 [×3], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×C10, C4×D8, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, D4⋊D5 [×4], C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C22×Dic5, D4×C10, C10.D8, C8×Dic5, D205C4, C5×D4⋊C4, D208C4, C2×D4⋊D5, D4×Dic5, Dic54D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, D8 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×D8, C4○D8, C4×D5 [×2], C22×D5, C4×D8, C2×C4×D5, D4×D5, D42D5, Dic54D4, D5×D8, SD163D5, Dic54D8

Smallest permutation representation of Dic54D8
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 68 6 63)(2 67 7 62)(3 66 8 61)(4 65 9 70)(5 64 10 69)(11 126 16 121)(12 125 17 130)(13 124 18 129)(14 123 19 128)(15 122 20 127)(21 55 26 60)(22 54 27 59)(23 53 28 58)(24 52 29 57)(25 51 30 56)(31 75 36 80)(32 74 37 79)(33 73 38 78)(34 72 39 77)(35 71 40 76)(41 85 46 90)(42 84 47 89)(43 83 48 88)(44 82 49 87)(45 81 50 86)(91 145 96 150)(92 144 97 149)(93 143 98 148)(94 142 99 147)(95 141 100 146)(101 135 106 140)(102 134 107 139)(103 133 108 138)(104 132 109 137)(105 131 110 136)(111 155 116 160)(112 154 117 159)(113 153 118 158)(114 152 119 157)(115 151 120 156)
(1 118 38 98 28 128 48 108)(2 117 39 97 29 127 49 107)(3 116 40 96 30 126 50 106)(4 115 31 95 21 125 41 105)(5 114 32 94 22 124 42 104)(6 113 33 93 23 123 43 103)(7 112 34 92 24 122 44 102)(8 111 35 91 25 121 45 101)(9 120 36 100 26 130 46 110)(10 119 37 99 27 129 47 109)(11 81 135 61 155 71 145 51)(12 90 136 70 156 80 146 60)(13 89 137 69 157 79 147 59)(14 88 138 68 158 78 148 58)(15 87 139 67 159 77 149 57)(16 86 140 66 160 76 150 56)(17 85 131 65 151 75 141 55)(18 84 132 64 152 74 142 54)(19 83 133 63 153 73 143 53)(20 82 134 62 154 72 144 52)
(1 98)(2 99)(3 100)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222222444444444444558888888810···1010101010202020202020202040···40
size11114420202244555510102020222222101010102···28888444488884···4

56 irreducible representations

dim1111111112222222224444
type++++++++++++++-++
imageC1C2C2C2C2C2C2C2C4D4D5D8C4○D4D10D10D10C4○D8C4×D5D42D5D4×D5D5×D8SD163D5
kernelDic54D8C10.D8C8×Dic5D205C4C5×D4⋊C4D208C4C2×D4⋊D5D4×Dic5D4⋊D5C2×Dic5D4⋊C4Dic5C20C4⋊C4C2×C8C2×D4C10D4C4C22C2C2
# reps1111111182242222482244

Matrix representation of Dic54D8 in GL4(𝔽41) generated by

40000
04000
00740
0010
,
9000
0900
002219
00919
,
01200
171700
00734
00134
,
01200
24000
00400
00040
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,7,1,0,0,40,0],[9,0,0,0,0,9,0,0,0,0,22,9,0,0,19,19],[0,17,0,0,12,17,0,0,0,0,7,1,0,0,34,34],[0,24,0,0,12,0,0,0,0,0,40,0,0,0,0,40] >;

Dic54D8 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_4D_8
% in TeX

G:=Group("Dic5:4D8");
// GroupNames label

G:=SmallGroup(320,383);
// by ID

G=gap.SmallGroup(320,383);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,135,100,570,297,136,12550]);
// Polycyclic

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

׿
×
𝔽