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G = D20.19D4order 320 = 26·5

2nd non-split extension by D20 of D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.19D4, C42.35D10, C4⋊C85D5, (C4×D20)⋊18C2, (C2×D40).4C2, (C2×C4).38D20, C4.131(D4×D5), (C2×C20).244D4, C20.340(C2×D4), (C2×C8).130D10, C53(D4.2D4), D205C412C2, C10.12(C4○D8), C20.44D48C2, C4.D2012C2, (C4×C20).70C22, (C2×C40).23C22, C20.329(C4○D4), C2.18(C8⋊D10), C10.39(C4⋊D4), C2.12(C4⋊D20), C10.15(C8⋊C22), (C2×C20).754C23, C4.45(Q82D5), (C2×D20).17C22, C22.117(C2×D20), C2.14(D407C2), C4⋊Dic5.274C22, (C2×Dic10).17C22, (C5×C4⋊C8)⋊7C2, (C2×C40⋊C2)⋊19C2, (C2×C10).137(C2×D4), (C2×C4).699(C22×D5), SmallGroup(320,471)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.19D4
C1C5C10C20C2×C20C2×D20C4×D20 — D20.19D4
C5C10C2×C20 — D20.19D4
C1C22C42C4⋊C8

Generators and relations for D20.19D4
 G = < a,b,c,d | a20=b2=c4=1, d2=a15, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a15b, dcd-1=a10c-1 >

Subgroups: 662 in 124 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], D4 [×5], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C22⋊C4 [×3], C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×7], C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], D20 [×3], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], D4.2D4, C40⋊C2 [×2], D40 [×2], C4⋊Dic5, D10⋊C4 [×3], C4×C20, C2×C40 [×2], C2×Dic10, C2×C4×D5, C2×D20 [×2], C20.44D4, D205C4, C5×C4⋊C8, C4×D20, C4.D20, C2×C40⋊C2, C2×D40, D20.19D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8⋊C22, D20 [×2], C22×D5, D4.2D4, C2×D20, D4×D5, Q82D5, C4⋊D20, D407C2, C8⋊D10, D20.19D4

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12222224444444455888810···1020···2020···2040···40
size1111202040222242020402244442···22···24···44···4

59 irreducible representations

dim111111112222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10C4○D8D20D407C2C8⋊C22D4×D5Q82D5C8⋊D10
kernelD20.19D4C20.44D4D205C4C5×C4⋊C8C4×D20C4.D20C2×C40⋊C2C2×D40D20C2×C20C4⋊C8C20C42C2×C8C10C2×C4C2C10C4C4C2
# reps1111111122222448161224

Matrix representation of D20.19D4 in GL4(𝔽41) generated by

303900
161400
00400
00040
,
38800
40300
003223
0099
,
32000
03200
004039
0011
,
203300
233800
003223
0009
G:=sub<GL(4,GF(41))| [30,16,0,0,39,14,0,0,0,0,40,0,0,0,0,40],[38,40,0,0,8,3,0,0,0,0,32,9,0,0,23,9],[32,0,0,0,0,32,0,0,0,0,40,1,0,0,39,1],[20,23,0,0,33,38,0,0,0,0,32,0,0,0,23,9] >;

D20.19D4 in GAP, Magma, Sage, TeX

D_{20}._{19}D_4
% in TeX

G:=Group("D20.19D4");
// GroupNames label

G:=SmallGroup(320,471);
// by ID

G=gap.SmallGroup(320,471);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^10*c^-1>;
// generators/relations

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