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G = C406D4order 320 = 26·5

6th semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C406D4, D103D8, (C2×D8)⋊5D5, (C10×D8)⋊6C2, C55(C87D4), C2.29(D5×D8), C202D44C2, C810(C5⋊D4), C405C423C2, C10.46(C2×D8), (C2×D4).64D10, C20.166(C2×D4), (C2×C8).238D10, C20.93(C4○D4), C10.34(C4○D8), D4⋊Dic529C2, (C2×C40).90C22, (C22×D5).89D4, C22.257(D4×D5), C4.28(D42D5), C2.18(D83D5), C2.16(C202D4), (C2×C20).434C23, (C2×Dic5).157D4, (D4×C10).83C22, C10.109(C4⋊D4), C4⋊Dic5.165C22, (D5×C2×C8)⋊3C2, C4.79(C2×C5⋊D4), (C2×C10).347(C2×D4), (C2×C4×D5).309C22, (C2×C4).524(C22×D5), (C2×C52C8).279C22, SmallGroup(320,784)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C406D4
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — C406D4
C5C10C2×C20 — C406D4
C1C22C2×C4C2×D8

Generators and relations for C406D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >

Subgroups: 566 in 134 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C5, C8 [×2], C8, C2×C4, C2×C4 [×5], D4 [×8], C23 [×3], D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8 [×3], D8 [×2], C22×C4, C2×D4 [×2], C2×D4 [×2], Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], D4⋊C4 [×2], C2.D8, C4⋊D4 [×2], C22×C8, C2×D8, C52C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C87D4, C8×D5 [×2], C2×C52C8, C4⋊Dic5 [×2], C23.D5 [×2], C2×C40, C5×D8 [×2], C2×C4×D5, C2×C5⋊D4 [×2], D4×C10 [×2], C405C4, D4⋊Dic5 [×2], D5×C2×C8, C202D4 [×2], C10×D8, C406D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C4○D8, C5⋊D4 [×2], C22×D5, C87D4, D4×D5, D42D5, C2×C5⋊D4, D5×D8, D83D5, C202D4, C406D4

Smallest permutation representation of C406D4
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)
(1 156 48 102)(2 155 49 101)(3 154 50 100)(4 153 51 99)(5 152 52 98)(6 151 53 97)(7 150 54 96)(8 149 55 95)(9 148 56 94)(10 147 57 93)(11 146 58 92)(12 145 59 91)(13 144 60 90)(14 143 61 89)(15 142 62 88)(16 141 63 87)(17 140 64 86)(18 139 65 85)(19 138 66 84)(20 137 67 83)(21 136 68 82)(22 135 69 81)(23 134 70 120)(24 133 71 119)(25 132 72 118)(26 131 73 117)(27 130 74 116)(28 129 75 115)(29 128 76 114)(30 127 77 113)(31 126 78 112)(32 125 79 111)(33 124 80 110)(34 123 41 109)(35 122 42 108)(36 121 43 107)(37 160 44 106)(38 159 45 105)(39 158 46 104)(40 157 47 103)
(2 10)(3 19)(4 28)(5 37)(7 15)(8 24)(9 33)(12 20)(13 29)(14 38)(17 25)(18 34)(22 30)(23 39)(27 35)(32 40)(41 65)(42 74)(44 52)(45 61)(46 70)(47 79)(49 57)(50 66)(51 75)(54 62)(55 71)(56 80)(59 67)(60 76)(64 72)(69 77)(81 127)(82 136)(83 145)(84 154)(85 123)(86 132)(87 141)(88 150)(89 159)(90 128)(91 137)(92 146)(93 155)(94 124)(95 133)(96 142)(97 151)(98 160)(99 129)(100 138)(101 147)(102 156)(103 125)(104 134)(105 143)(106 152)(107 121)(108 130)(109 139)(110 148)(111 157)(112 126)(113 135)(114 144)(115 153)(116 122)(117 131)(118 140)(119 149)(120 158)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B8C8D8E8F8G8H10A···10F10G···10N20A20B20C20D40A···40H
order12222222444444558888888810···1010···102020202040···40
size11118810102210104040222222101010102···28···844444···4

50 irreducible representations

dim11111122222222224444
type+++++++++++++-++-
imageC1C2C2C2C2C2D4D4D4D5C4○D4D8D10D10C4○D8C5⋊D4D42D5D4×D5D5×D8D83D5
kernelC406D4C405C4D4⋊Dic5D5×C2×C8C202D4C10×D8C40C2×Dic5C22×D5C2×D8C20D10C2×C8C2×D4C10C8C4C22C2C2
# reps11212121122424482244

Matrix representation of C406D4 in GL4(𝔽41) generated by

35700
35000
0030
002414
,
211700
152000
0019
00040
,
6100
63500
0010
0001
G:=sub<GL(4,GF(41))| [35,35,0,0,7,0,0,0,0,0,3,24,0,0,0,14],[21,15,0,0,17,20,0,0,0,0,1,0,0,0,9,40],[6,6,0,0,1,35,0,0,0,0,1,0,0,0,0,1] >;

C406D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_6D_4
% in TeX

G:=Group("C40:6D4");
// GroupNames label

G:=SmallGroup(320,784);
// by ID

G=gap.SmallGroup(320,784);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

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