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G = D43D20order 320 = 26·5

2nd semidirect product of D4 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D43D20, Dic103D4, (C5×D4)⋊2D4, C20.2(C2×D4), C4.86(D4×D5), C4.3(C2×D20), C4⋊D203C2, C52(D4⋊D4), C4⋊C4.12D10, D4⋊C413D5, C10.Q166C2, (C2×C8).117D10, D101C812C2, C10.21C22≀C2, (C2×D4).136D10, C10.42(C4○D8), (C22×D5).21D4, C22.179(D4×D5), C2.17(D8⋊D5), C10.35(C8⋊C22), (C2×C40).128C22, (C2×C20).221C23, (C2×Dic5).198D4, (D4×C10).42C22, (C2×D20).55C22, C2.24(C22⋊D20), C2.12(SD163D5), (C2×Dic10).63C22, (C2×D4⋊D5)⋊4C2, (C2×D42D5)⋊1C2, (C2×C40⋊C2)⋊15C2, (C5×D4⋊C4)⋊13C2, (C2×C4×D5).17C22, (C2×C10).234(C2×D4), (C5×C4⋊C4).22C22, (C2×C52C8).19C22, (C2×C4).328(C22×D5), SmallGroup(320,408)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D43D20
C1C5C10C2×C10C2×C20C2×C4×D5C2×D42D5 — D43D20
C5C10C2×C20 — D43D20
C1C22C2×C4D4⋊C4

Generators and relations for D43D20
 G = < a,b,c,d | a4=b2=c20=d2=1, bab=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 782 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4, C2×C4 [×9], D4 [×2], D4 [×9], Q8 [×3], C23 [×3], D5 [×2], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×3], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×4], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C52C8, C40, Dic10 [×2], Dic10, C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5 [×5], C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×D5, C22×C10, D4⋊D4, C40⋊C2 [×2], C2×C52C8, D10⋊C4, D4⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D42D5 [×4], C22×Dic5, C2×C5⋊D4, D4×C10, C10.Q16, D101C8, C5×D4⋊C4, C4⋊D20, C2×C40⋊C2, C2×D4⋊D5, C2×D42D5, D43D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, D20 [×2], C22×D5, D4⋊D4, C2×D20, D4×D5 [×2], C22⋊D20, D8⋊D5, SD163D5, D43D20

Smallest permutation representation of D43D20
On 160 points
Generators in S160
(1 109 151 35)(2 36 152 110)(3 111 153 37)(4 38 154 112)(5 113 155 39)(6 40 156 114)(7 115 157 21)(8 22 158 116)(9 117 159 23)(10 24 160 118)(11 119 141 25)(12 26 142 120)(13 101 143 27)(14 28 144 102)(15 103 145 29)(16 30 146 104)(17 105 147 31)(18 32 148 106)(19 107 149 33)(20 34 150 108)(41 99 131 79)(42 80 132 100)(43 81 133 61)(44 62 134 82)(45 83 135 63)(46 64 136 84)(47 85 137 65)(48 66 138 86)(49 87 139 67)(50 68 140 88)(51 89 121 69)(52 70 122 90)(53 91 123 71)(54 72 124 92)(55 93 125 73)(56 74 126 94)(57 95 127 75)(58 76 128 96)(59 97 129 77)(60 78 130 98)
(1 91)(2 124)(3 93)(4 126)(5 95)(6 128)(7 97)(8 130)(9 99)(10 132)(11 81)(12 134)(13 83)(14 136)(15 85)(16 138)(17 87)(18 140)(19 89)(20 122)(21 129)(22 78)(23 131)(24 80)(25 133)(26 62)(27 135)(28 64)(29 137)(30 66)(31 139)(32 68)(33 121)(34 70)(35 123)(36 72)(37 125)(38 74)(39 127)(40 76)(41 117)(42 160)(43 119)(44 142)(45 101)(46 144)(47 103)(48 146)(49 105)(50 148)(51 107)(52 150)(53 109)(54 152)(55 111)(56 154)(57 113)(58 156)(59 115)(60 158)(61 141)(63 143)(65 145)(67 147)(69 149)(71 151)(73 153)(75 155)(77 157)(79 159)(82 120)(84 102)(86 104)(88 106)(90 108)(92 110)(94 112)(96 114)(98 116)(100 118)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 117)(22 116)(23 115)(24 114)(25 113)(26 112)(27 111)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 120)(39 119)(40 118)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 100)(59 99)(60 98)(61 127)(62 126)(63 125)(64 124)(65 123)(66 122)(67 121)(68 140)(69 139)(70 138)(71 137)(72 136)(73 135)(74 134)(75 133)(76 132)(77 131)(78 130)(79 129)(80 128)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222222444444455888810···1010101010202020202020202040···40
size111144204022810102020224420202···28888444488884···4

47 irreducible representations

dim11111111222222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10D10C4○D8D20C8⋊C22D4×D5D4×D5D8⋊D5SD163D5
kernelD43D20C10.Q16D101C8C5×D4⋊C4C4⋊D20C2×C40⋊C2C2×D4⋊D5C2×D42D5Dic10C2×Dic5C5×D4C22×D5D4⋊C4C4⋊C4C2×C8C2×D4C10D4C10C4C22C2C2
# reps11111111212122224812244

Matrix representation of D43D20 in GL4(𝔽41) generated by

1000
0100
00121
003740
,
40000
04000
00035
00340
,
23000
271600
00925
00032
,
40000
5100
00400
0041
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,37,0,0,21,40],[40,0,0,0,0,40,0,0,0,0,0,34,0,0,35,0],[2,27,0,0,30,16,0,0,0,0,9,0,0,0,25,32],[40,5,0,0,0,1,0,0,0,0,40,4,0,0,0,1] >;

D43D20 in GAP, Magma, Sage, TeX

D_4\rtimes_3D_{20}
% in TeX

G:=Group("D4:3D20");
// GroupNames label

G:=SmallGroup(320,408);
// by ID

G=gap.SmallGroup(320,408);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,58,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽