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G = C2×C20.C8order 320 = 26·5

Direct product of C2 and C20.C8

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

Aliases: C2×C20.C8, C102M5(2), C5⋊C163C22, (C2×C20).5C8, C53(C2×M5(2)), C20.43(C2×C8), C23.3(C5⋊C8), (C22×C10).6C8, (C22×C4).18F5, C4.55(C22×F5), C10.17(C22×C8), C20.95(C22×C4), (C22×C20).27C4, C52C8.40C23, C4.9(C2×C5⋊C8), (C2×C5⋊C16)⋊8C2, (C2×C4).6(C5⋊C8), C22.5(C2×C5⋊C8), C2.3(C22×C5⋊C8), (C2×C52C8).28C4, (C2×C10).32(C2×C8), C52C8.56(C2×C4), (C2×C4).168(C2×F5), (C2×C20).149(C2×C4), (C22×C52C8).21C2, (C2×C52C8).339C22, SmallGroup(320,1081)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C20.C8
C1C5C10C20C52C8C5⋊C16C2×C5⋊C16 — C2×C20.C8
C5C10 — C2×C20.C8
C1C2×C4C22×C4

Generators and relations for C2×C20.C8
 G = < a,b,c | a2=b20=1, c8=b10, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 186 in 90 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C16 [×4], C2×C8 [×6], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C16 [×2], M5(2) [×4], C22×C8, C52C8 [×2], C52C8 [×2], C2×C20 [×2], C2×C20 [×4], C22×C10, C2×M5(2), C5⋊C16 [×4], C2×C52C8 [×2], C2×C52C8 [×4], C22×C20, C2×C5⋊C16 [×2], C20.C8 [×4], C22×C52C8, C2×C20.C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, M5(2) [×2], C22×C8, C5⋊C8 [×4], C2×F5 [×3], C2×M5(2), C2×C5⋊C8 [×6], C22×F5, C20.C8 [×2], C22×C5⋊C8, C2×C20.C8

Smallest permutation representation of C2×C20.C8
On 160 points
Generators in S160
(1 150)(2 151)(3 152)(4 153)(5 154)(6 155)(7 156)(8 157)(9 158)(10 159)(11 160)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 56)(22 57)(23 58)(24 59)(25 60)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 101)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)(93 138)(94 139)(95 140)(96 121)(97 122)(98 123)(99 124)(100 125)
(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 92 30 65 155 132 50 101 11 82 40 75 145 122 60 111)(2 99 39 68 156 139 59 104 12 89 29 78 146 129 49 114)(3 86 28 71 157 126 48 107 13 96 38 61 147 136 58 117)(4 93 37 74 158 133 57 110 14 83 27 64 148 123 47 120)(5 100 26 77 159 140 46 113 15 90 36 67 149 130 56 103)(6 87 35 80 160 127 55 116 16 97 25 70 150 137 45 106)(7 94 24 63 141 134 44 119 17 84 34 73 151 124 54 109)(8 81 33 66 142 121 53 102 18 91 23 76 152 131 43 112)(9 88 22 69 143 128 42 105 19 98 32 79 153 138 52 115)(10 95 31 72 144 135 51 108 20 85 21 62 154 125 41 118)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F 5 8A···8H8I8J8K8L10A···10G16A···16P20A···20H
order12222244444458···8888810···1016···1620···20
size11112211112245···5101010104···410···104···4

56 irreducible representations

dim11111111244444
type+++++-+-
imageC1C2C2C2C4C4C8C8M5(2)F5C5⋊C8C2×F5C5⋊C8C20.C8
kernelC2×C20.C8C2×C5⋊C16C20.C8C22×C52C8C2×C52C8C22×C20C2×C20C22×C10C10C22×C4C2×C4C2×C4C23C2
# reps124162124813318

Matrix representation of C2×C20.C8 in GL6(𝔽241)

100000
010000
00240000
00024000
00002400
00000240
,
17700000
236640000
001316400
00177000
001161170110
00587195195
,
177960000
43640000
0000190240
00646421
00831851770
002072151310

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[177,236,0,0,0,0,0,64,0,0,0,0,0,0,131,177,116,58,0,0,64,0,117,7,0,0,0,0,0,195,0,0,0,0,110,195],[177,43,0,0,0,0,96,64,0,0,0,0,0,0,0,64,83,207,0,0,0,64,185,215,0,0,190,2,177,131,0,0,240,1,0,0] >;

C2×C20.C8 in GAP, Magma, Sage, TeX

C_2\times C_{20}.C_8
% in TeX

G:=Group("C2xC20.C8");
// GroupNames label

G:=SmallGroup(320,1081);
// by ID

G=gap.SmallGroup(320,1081);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,80,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^2=b^20=1,c^8=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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