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G = D20.47D4order 320 = 26·5

3rd non-split extension by D20 of D4 acting through Inn(D20)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.47D4, D8.10D10, C40.39C23, C20.18C24, Q16.12D10, SD16.2D10, Dic10.47D4, Dic10.12C23, Dic20.16C22, C4○D86D5, C53(Q8○D8), (D5×Q16)⋊7C2, C5⋊D4.3D4, D83D57C2, C4.145(D4×D5), D4.D5.C22, C4○D4.13D10, D10.54(C2×D4), (C2×C8).106D10, SD16⋊D56C2, C20.351(C2×D4), C52C8.9C23, (C8×D5).8C22, C4.18(C23×D5), C22.10(D4×D5), C8.18(C22×D5), (C2×Dic20)⋊23C2, D4.9D108C2, D20.3C48C2, (Q8×D5).2C22, Dic5.60(C2×D4), (C5×D4).12C23, D4.12(C22×D5), (C4×D5).11C23, (C5×D8).10C22, C8⋊D5.2C22, D4.10D106C2, Q8.12(C22×D5), (C5×Q8).12C23, C5⋊Q16.2C22, (C2×C20).535C23, (C2×C40).106C22, C4○D20.56C22, D42D5.2C22, C10.119(C22×D4), (C5×Q16).12C22, (C5×SD16).2C22, C4.Dic5.49C22, (C2×Dic10).206C22, C2.92(C2×D4×D5), (C5×C4○D8)⋊6C2, (C2×C10).15(C2×D4), (C5×C4○D4).23C22, (C2×C4).234(C22×D5), SmallGroup(320,1443)

Series: Derived Chief Lower central Upper central

C1C20 — D20.47D4
C1C5C10C20C4×D5C4○D20D4.10D10 — D20.47D4
C5C10C20 — D20.47D4
C1C2C2×C4C4○D8

Generators and relations for D20.47D4
 G = < a,b,c,d | a8=b2=c10=1, d2=a4, bab=a-1, ac=ca, ad=da, cbc-1=a4b, bd=db, dcd-1=a4c-1 >

Subgroups: 854 in 248 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4 [×2], D4 [×9], Q8 [×2], Q8 [×11], D5 [×2], C10, C10 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, SD16 [×2], SD16 [×4], Q16, Q16 [×8], C2×Q8 [×8], C4○D4 [×2], C4○D4 [×11], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], C2×C10, C2×C10 [×2], C8○D4, C2×Q16 [×3], C4○D8, C4○D8 [×2], C8.C22 [×6], 2- 1+4 [×2], C52C8 [×2], C40 [×2], Dic10, Dic10 [×4], Dic10 [×6], C4×D5 [×2], C4×D5 [×4], D20, C2×Dic5 [×6], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], Q8○D8, C8×D5 [×2], C8⋊D5 [×2], Dic20 [×4], C4.Dic5, D4.D5 [×4], C5⋊Q16 [×4], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C2×Dic10 [×2], C2×Dic10 [×2], C4○D20, C4○D20 [×2], D42D5 [×4], D42D5 [×4], Q8×D5 [×4], C5×C4○D4 [×2], D20.3C4, C2×Dic20, D83D5 [×2], SD16⋊D5 [×4], D5×Q16 [×2], D4.9D10 [×2], C5×C4○D8, D4.10D10 [×2], D20.47D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], Q8○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D20.47D4

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E10A10B10C10D10E10F10G10H20A20B20C20D20E20F20G20H20I20J40A···40H
order12222224444444444558888810101010101010102020202020202020202040···40
size11244101022441010202020202222420202244888822224488884···4

50 irreducible representations

dim1111111112222222224444
type++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10Q8○D8D4×D5D4×D5D20.47D4
kernelD20.47D4D20.3C4C2×Dic20D83D5SD16⋊D5D5×Q16D4.9D10C5×C4○D8D4.10D10Dic10D20C5⋊D4C4○D8C2×C8D8SD16Q16C4○D4C5C4C22C1
# reps1112422121122224242228

Matrix representation of D20.47D4 in GL4(𝔽41) generated by

00338
00330
036240
536024
,
5293128
12361018
6183412
2424277
,
39162315
2525638
773225
114527
,
39132115
2521838
7351425
1332527
G:=sub<GL(4,GF(41))| [0,0,0,5,0,0,36,36,33,33,24,0,8,0,0,24],[5,12,6,24,29,36,18,24,31,10,34,27,28,18,12,7],[39,25,7,1,16,25,7,14,23,6,32,5,15,38,25,27],[39,25,7,1,13,2,35,33,21,18,14,25,15,38,25,27] >;

D20.47D4 in GAP, Magma, Sage, TeX

D_{20}._{47}D_4
% in TeX

G:=Group("D20.47D4");
// GroupNames label

G:=SmallGroup(320,1443);
// by ID

G=gap.SmallGroup(320,1443);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,184,570,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations

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