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G = C8⋊D20order 320 = 26·5

1st semidirect product of C8 and D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C401D4, C81D20, C42.18D10, C8⋊C43D5, C51(C83D4), (C2×D40)⋊10C2, C204D42C2, (C2×C4).25D20, (C2×C8).55D10, C4.34(C2×D20), (C2×C20).36D4, C20.277(C2×D4), C4.D202C2, (C4×C20).3C22, C2.7(C8⋊D10), C2.9(C204D4), C10.7(C41D4), C10.4(C8⋊C22), (C2×C40).56C22, (C2×D20).8C22, C22.97(C2×D20), (C2×C20).733C23, (C2×Dic10).9C22, (C5×C8⋊C4)⋊4C2, (C2×C40⋊C2)⋊1C2, (C2×C10).116(C2×D4), (C2×C4).677(C22×D5), SmallGroup(320,339)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C8⋊D20
C1C5C10C20C2×C20C2×D20C204D4 — C8⋊D20
C5C10C2×C20 — C8⋊D20
C1C22C42C8⋊C4

Generators and relations for C8⋊D20
 G = < a,b,c | a8=b20=c2=1, bab-1=a5, cac=a3, cbc=b-1 >

Subgroups: 878 in 144 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], D5 [×3], C10, C10 [×2], C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×4], SD16 [×4], C2×D4 [×5], C2×Q8, Dic5, C20 [×2], C20 [×2], D10 [×9], C2×C10, C8⋊C4, C4.4D4, C41D4, C2×D8 [×2], C2×SD16 [×2], C40 [×4], Dic10 [×2], D20 [×10], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5 [×3], C83D4, C40⋊C2 [×4], D40 [×4], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×Dic10, C2×D20, C2×D20 [×2], C2×D20 [×2], C5×C8⋊C4, C204D4, C4.D20, C2×C40⋊C2 [×2], C2×D40 [×2], C8⋊D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C8⋊C22 [×2], D20 [×6], C22×D5, C83D4, C2×D20 [×3], C204D4, C8⋊D10 [×2], C8⋊D20

Smallest permutation representation of C8⋊D20
On 160 points
Generators in S160
(1 159 104 45 30 73 98 124)(2 74 105 125 31 160 99 46)(3 141 106 47 32 75 100 126)(4 76 107 127 33 142 81 48)(5 143 108 49 34 77 82 128)(6 78 109 129 35 144 83 50)(7 145 110 51 36 79 84 130)(8 80 111 131 37 146 85 52)(9 147 112 53 38 61 86 132)(10 62 113 133 39 148 87 54)(11 149 114 55 40 63 88 134)(12 64 115 135 21 150 89 56)(13 151 116 57 22 65 90 136)(14 66 117 137 23 152 91 58)(15 153 118 59 24 67 92 138)(16 68 119 139 25 154 93 60)(17 155 120 41 26 69 94 140)(18 70 101 121 27 156 95 42)(19 157 102 43 28 71 96 122)(20 72 103 123 29 158 97 44)
(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)
(1 123)(2 122)(3 121)(4 140)(5 139)(6 138)(7 137)(8 136)(9 135)(10 134)(11 133)(12 132)(13 131)(14 130)(15 129)(16 128)(17 127)(18 126)(19 125)(20 124)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(61 115)(62 114)(63 113)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 105)(72 104)(73 103)(74 102)(75 101)(76 120)(77 119)(78 118)(79 117)(80 116)(81 155)(82 154)(83 153)(84 152)(85 151)(86 150)(87 149)(88 148)(89 147)(90 146)(91 145)(92 144)(93 143)(94 142)(95 141)(96 160)(97 159)(98 158)(99 157)(100 156)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12222224444455888810···1020···2020···2040···40
size11114040402244402244442···22···24···44···4

56 irreducible representations

dim111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10D20D20C8⋊C22C8⋊D10
kernelC8⋊D20C5×C8⋊C4C204D4C4.D20C2×C40⋊C2C2×D40C40C2×C20C8⋊C4C42C2×C8C8C2×C4C10C2
# reps1111224222416828

Matrix representation of C8⋊D20 in GL6(𝔽41)

2280000
13390000
00286627
0030391235
00301335
0003112
,
16390000
2280000
0035331334
0013467
0023368
0011407
,
160000
0400000
0000400
000051
0040000
005100

G:=sub<GL(6,GF(41))| [2,13,0,0,0,0,28,39,0,0,0,0,0,0,28,30,3,0,0,0,6,39,0,3,0,0,6,12,13,11,0,0,27,35,35,2],[16,2,0,0,0,0,39,28,0,0,0,0,0,0,35,1,2,1,0,0,33,34,33,1,0,0,13,6,6,40,0,0,34,7,8,7],[1,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,40,5,0,0,0,0,0,1,0,0] >;

C8⋊D20 in GAP, Magma, Sage, TeX

C_8\rtimes D_{20}
% in TeX

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

G:=SmallGroup(320,339);
// by ID

G=gap.SmallGroup(320,339);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations

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