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G = D203C8order 320 = 26·5

1st semidirect product of D20 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D203C8, C20.35D8, C4.17D40, C20.29SD16, C42.249D10, C20.36M4(2), (C4×C8)⋊1D5, (C4×C40)⋊1C2, C53(D4⋊C8), C4.7(C8×D5), C203C81C2, C10.16C4≀C2, C20.51(C2×C8), (C4×D20).1C2, (C2×D20).16C4, (C2×C20).436D4, (C2×C4).161D20, C4.5(C8⋊D5), C4⋊Dic5.17C4, C4.16(C40⋊C2), C2.1(D205C4), C2.4(D101C8), C2.2(D204C4), C10.17(C22⋊C8), (C4×C20).320C22, C10.25(D4⋊C4), C22.31(D10⋊C4), (C2×C4).96(C4×D5), (C2×C20).389(C2×C4), (C2×C4).205(C5⋊D4), (C2×C10).100(C22⋊C4), SmallGroup(320,17)

Series: Derived Chief Lower central Upper central

C1C20 — D203C8
C1C5C10C2×C10C2×C20C4×C20C4×D20 — D203C8
C5C10C20 — D203C8
C1C2×C4C42C4×C8

Generators and relations for D203C8
 G = < a,b,c | a20=b2=c8=1, bab=a-1, ac=ca, cbc-1=a15b >

Subgroups: 374 in 82 conjugacy classes, 35 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C22, C22 [×4], C5, C8 [×3], C2×C4 [×3], C2×C4 [×3], D4 [×3], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C22×C4, C2×D4, Dic5, C20 [×4], C20, D10 [×4], C2×C10, C4×C8, C4⋊C8, C4×D4, C52C8, C40 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C22×D5, D4⋊C8, C2×C52C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C203C8, C4×C40, C4×D20, D203C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), D8, SD16, D10, C22⋊C8, D4⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, D4⋊C8, C8×D5, C8⋊D5, C40⋊C2, D40, D10⋊C4, D204C4, D101C8, D205C4, D203C8

Smallest permutation representation of D203C8
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)
(1 130)(2 129)(3 128)(4 127)(5 126)(6 125)(7 124)(8 123)(9 122)(10 121)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 45)(22 44)(23 43)(24 42)(25 41)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 120)(68 119)(69 118)(70 117)(71 116)(72 115)(73 114)(74 113)(75 112)(76 111)(77 110)(78 109)(79 108)(80 107)(81 149)(82 148)(83 147)(84 146)(85 145)(86 144)(87 143)(88 142)(89 141)(90 160)(91 159)(92 158)(93 157)(94 156)(95 155)(96 154)(97 153)(98 152)(99 151)(100 150)
(1 24 69 90 121 48 119 156)(2 25 70 91 122 49 120 157)(3 26 71 92 123 50 101 158)(4 27 72 93 124 51 102 159)(5 28 73 94 125 52 103 160)(6 29 74 95 126 53 104 141)(7 30 75 96 127 54 105 142)(8 31 76 97 128 55 106 143)(9 32 77 98 129 56 107 144)(10 33 78 99 130 57 108 145)(11 34 79 100 131 58 109 146)(12 35 80 81 132 59 110 147)(13 36 61 82 133 60 111 148)(14 37 62 83 134 41 112 149)(15 38 63 84 135 42 113 150)(16 39 64 85 136 43 114 151)(17 40 65 86 137 44 115 152)(18 21 66 87 138 45 116 153)(19 22 67 88 139 46 117 154)(20 23 68 89 140 47 118 155)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A···8H8I8J8K8L10A···10F20A···20X40A···40AF
order1222224444444444558···8888810···1020···2040···40
size11112020111122222020222···2202020202···22···22···2

92 irreducible representations

dim1111111222222222222222
type++++++++++
imageC1C2C2C2C4C4C8D4D5M4(2)D8SD16D10C4≀C2C4×D5D20C5⋊D4C8×D5C8⋊D5C40⋊C2D40D204C4
kernelD203C8C203C8C4×C40C4×D20C4⋊Dic5C2×D20D20C2×C20C4×C8C20C20C20C42C10C2×C4C2×C4C2×C4C4C4C4C4C2
# reps11112282222224444888816

Matrix representation of D203C8 in GL4(𝔽41) generated by

0100
403400
0001
00400
,
04000
40000
0001
0010
,
38000
03800
002912
002929
G:=sub<GL(4,GF(41))| [0,40,0,0,1,34,0,0,0,0,0,40,0,0,1,0],[0,40,0,0,40,0,0,0,0,0,0,1,0,0,1,0],[38,0,0,0,0,38,0,0,0,0,29,29,0,0,12,29] >;

D203C8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_3C_8
% in TeX

G:=Group("D20:3C8");
// GroupNames label

G:=SmallGroup(320,17);
// by ID

G=gap.SmallGroup(320,17);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,100,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations

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