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G = C405C4⋊C2order 320 = 26·5

9th semidirect product of C405C4 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C405C49C2, D4⋊C47D5, (C2×C8).11D10, D101C87C2, C4⋊C4.138D10, (C2×D4).30D10, C202D4.7C2, C20.Q87C2, C10.25(C4○D8), C4.54(C4○D20), D4⋊Dic510C2, (C2×C40).11C22, (C22×D5).24D4, C22.182(D4×D5), C20.152(C4○D4), C2.11(D83D5), C4.81(D42D5), C2.12(D40⋊C2), C10.56(C8⋊C22), (C2×C20).224C23, (C2×Dic5).200D4, (D4×C10).45C22, C53(C23.19D4), C4⋊Dic5.76C22, C2.15(D10.12D4), C10.23(C22.D4), C4⋊C47D54C2, (C5×D4⋊C4)⋊7C2, (C2×C4×D5).20C22, (C2×C10).237(C2×D4), (C5×C4⋊C4).25C22, (C2×C52C8).22C22, (C2×C4).331(C22×D5), SmallGroup(320,411)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C405C4⋊C2
C1C5C10C2×C10C2×C20C2×C4×D5C4⋊C47D5 — C405C4⋊C2
C5C10C2×C20 — C405C4⋊C2
C1C22C2×C4D4⋊C4

Generators and relations for C405C4⋊C2
 G = < a,b,c | a40=b4=c2=1, bab-1=a-1, cac=a11b2, cbc=b-1 >

Subgroups: 446 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×4], C23 [×2], D5, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C52C8, C40, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C23.19D4, C2×C52C8, C4×Dic5, C4⋊Dic5 [×2], D10⋊C4, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C5⋊D4, D4×C10, C20.Q8, C405C4, D101C8, D4⋊Dic5, C5×D4⋊C4, C4⋊C47D5, C202D4, C405C4⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8⋊C22, C22×D5, C23.19D4, C4○D20, D4×D5, D42D5, D10.12D4, D83D5, D40⋊C2, C405C4⋊C2

Smallest permutation representation of C405C4⋊C2
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 51 122 117)(2 50 123 116)(3 49 124 115)(4 48 125 114)(5 47 126 113)(6 46 127 112)(7 45 128 111)(8 44 129 110)(9 43 130 109)(10 42 131 108)(11 41 132 107)(12 80 133 106)(13 79 134 105)(14 78 135 104)(15 77 136 103)(16 76 137 102)(17 75 138 101)(18 74 139 100)(19 73 140 99)(20 72 141 98)(21 71 142 97)(22 70 143 96)(23 69 144 95)(24 68 145 94)(25 67 146 93)(26 66 147 92)(27 65 148 91)(28 64 149 90)(29 63 150 89)(30 62 151 88)(31 61 152 87)(32 60 153 86)(33 59 154 85)(34 58 155 84)(35 57 156 83)(36 56 157 82)(37 55 158 81)(38 54 159 120)(39 53 160 119)(40 52 121 118)
(1 51)(2 88)(3 73)(4 110)(5 55)(6 92)(7 77)(8 114)(9 59)(10 96)(11 41)(12 118)(13 63)(14 100)(15 45)(16 82)(17 67)(18 104)(19 49)(20 86)(21 71)(22 108)(23 53)(24 90)(25 75)(26 112)(27 57)(28 94)(29 79)(30 116)(31 61)(32 98)(33 43)(34 120)(35 65)(36 102)(37 47)(38 84)(39 69)(40 106)(42 143)(44 125)(46 147)(48 129)(50 151)(52 133)(54 155)(56 137)(58 159)(60 141)(62 123)(64 145)(66 127)(68 149)(70 131)(72 153)(74 135)(76 157)(78 139)(80 121)(81 126)(83 148)(85 130)(87 152)(89 134)(91 156)(93 138)(95 160)(97 142)(99 124)(101 146)(103 128)(105 150)(107 132)(109 154)(111 136)(113 158)(115 140)(117 122)(119 144)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444455888810···1010101010202020202020202040···40
size111182022441010202040224420202···28888444488884···4

47 irreducible representations

dim1111111122222222244444
type+++++++++++++++-+-+
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C4○D20C8⋊C22D42D5D4×D5D83D5D40⋊C2
kernelC405C4⋊C2C20.Q8C405C4D101C8D4⋊Dic5C5×D4⋊C4C4⋊C47D5C202D4C2×Dic5C22×D5D4⋊C4C20C4⋊C4C2×C8C2×D4C10C4C10C4C22C2C2
# reps1111111111242224812244

Matrix representation of C405C4⋊C2 in GL4(𝔽41) generated by

3000
81400
003916
002525
,
29400
151200
002020
002321
,
123700
52900
00186
003523
G:=sub<GL(4,GF(41))| [3,8,0,0,0,14,0,0,0,0,39,25,0,0,16,25],[29,15,0,0,4,12,0,0,0,0,20,23,0,0,20,21],[12,5,0,0,37,29,0,0,0,0,18,35,0,0,6,23] >;

C405C4⋊C2 in GAP, Magma, Sage, TeX

C_{40}\rtimes_5C_4\rtimes C_2
% in TeX

G:=Group("C40:5C4:C2");
// GroupNames label

G:=SmallGroup(320,411);
// by ID

G=gap.SmallGroup(320,411);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,926,219,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^11*b^2,c*b*c=b^-1>;
// generators/relations

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