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G = D4.2D20order 320 = 26·5

2nd non-split extension by D4 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.2D20, C2010SD16, C42.52D10, (C4×D4).9D5, C43(D4.D5), C203C825C2, (C5×D4).19D4, C4.15(C2×D20), C20.19(C2×D4), (C2×C20).62D4, C4⋊C4.247D10, C202Q816C2, (D4×C20).10C2, C53(D4.D4), (C2×D4).194D10, C20.54(C4○D4), C4.11(C4○D20), C10.Q1631C2, (C4×C20).90C22, C10.53(C2×SD16), C10.66(C4⋊D4), C2.14(C207D4), (C2×C20).341C23, C2.8(D4.9D10), (D4×C10).236C22, C10.109(C8.C22), (C2×Dic10).104C22, C2.7(C2×D4.D5), (C2×D4.D5).5C2, (C2×C10).472(C2×D4), (C2×C4).246(C5⋊D4), (C5×C4⋊C4).278C22, (C2×C52C8).96C22, (C2×C4).441(C22×D5), C22.151(C2×C5⋊D4), SmallGroup(320,646)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.2D20
C1C5C10C20C2×C20C2×Dic10C202Q8 — D4.2D20
C5C10C2×C20 — D4.2D20
C1C22C42C4×D4

Generators and relations for D4.2D20
 G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 406 in 120 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×4], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8 [×2], Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C52C8 [×2], Dic10 [×4], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, D4.D4, C2×C52C8 [×2], C4⋊Dic5 [×2], D4.D5 [×4], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10 [×2], C22×C20, D4×C10, C203C8, C10.Q16 [×2], C202Q8, C2×D4.D5 [×2], D4×C20, D4.2D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8.C22, D20 [×2], C5⋊D4 [×2], C22×D5, D4.D4, D4.D5 [×2], C2×D20, C4○D20, C2×C5⋊D4, C207D4, C2×D4.D5, D4.9D10, D4.2D20

Smallest permutation representation of D4.2D20
On 160 points
Generators in S160
(1 105 56 152)(2 106 57 153)(3 107 58 154)(4 108 59 155)(5 109 60 156)(6 110 41 157)(7 111 42 158)(8 112 43 159)(9 113 44 160)(10 114 45 141)(11 115 46 142)(12 116 47 143)(13 117 48 144)(14 118 49 145)(15 119 50 146)(16 120 51 147)(17 101 52 148)(18 102 53 149)(19 103 54 150)(20 104 55 151)(21 129 83 75)(22 130 84 76)(23 131 85 77)(24 132 86 78)(25 133 87 79)(26 134 88 80)(27 135 89 61)(28 136 90 62)(29 137 91 63)(30 138 92 64)(31 139 93 65)(32 140 94 66)(33 121 95 67)(34 122 96 68)(35 123 97 69)(36 124 98 70)(37 125 99 71)(38 126 100 72)(39 127 81 73)(40 128 82 74)
(1 142)(2 143)(3 144)(4 145)(5 146)(6 147)(7 148)(8 149)(9 150)(10 151)(11 152)(12 153)(13 154)(14 155)(15 156)(16 157)(17 158)(18 159)(19 160)(20 141)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 120)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)
(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 33 56 95)(2 32 57 94)(3 31 58 93)(4 30 59 92)(5 29 60 91)(6 28 41 90)(7 27 42 89)(8 26 43 88)(9 25 44 87)(10 24 45 86)(11 23 46 85)(12 22 47 84)(13 21 48 83)(14 40 49 82)(15 39 50 81)(16 38 51 100)(17 37 52 99)(18 36 53 98)(19 35 54 97)(20 34 55 96)(61 158 135 111)(62 157 136 110)(63 156 137 109)(64 155 138 108)(65 154 139 107)(66 153 140 106)(67 152 121 105)(68 151 122 104)(69 150 123 103)(70 149 124 102)(71 148 125 101)(72 147 126 120)(73 146 127 119)(74 145 128 118)(75 144 129 117)(76 143 130 116)(77 142 131 115)(78 141 132 114)(79 160 133 113)(80 159 134 112)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G···10N20A···20H20I···20X
order12222244444444455888810···1010···1020···2020···20
size1111442222444404022202020202···24···42···24···4

59 irreducible representations

dim11111122222222222444
type+++++++++++++---
imageC1C2C2C2C2C2D4D4D5SD16C4○D4D10D10D10C5⋊D4D20C4○D20C8.C22D4.D5D4.9D10
kernelD4.2D20C203C8C10.Q16C202Q8C2×D4.D5D4×C20C2×C20C5×D4C4×D4C20C20C42C4⋊C4C2×D4C2×C4D4C4C10C4C2
# reps11212122242222888144

Matrix representation of D4.2D20 in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000222
00002439
,
100000
8400000
0040000
0039100
0000222
00003939
,
3200000
1090000
0016000
00391800
0000400
0000040
,
23250000
33180000
00281300
0061300
0000152
00001026

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,24,0,0,0,0,22,39],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,40,39,0,0,0,0,0,1,0,0,0,0,0,0,2,39,0,0,0,0,22,39],[32,10,0,0,0,0,0,9,0,0,0,0,0,0,16,39,0,0,0,0,0,18,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[23,33,0,0,0,0,25,18,0,0,0,0,0,0,28,6,0,0,0,0,13,13,0,0,0,0,0,0,15,10,0,0,0,0,2,26] >;

D4.2D20 in GAP, Magma, Sage, TeX

D_4._2D_{20}
% in TeX

G:=Group("D4.2D20");
// GroupNames label

G:=SmallGroup(320,646);
// by ID

G=gap.SmallGroup(320,646);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,1123,297,136,12550]);
// Polycyclic

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

׿
×
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