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G = D4.5D20order 320 = 26·5

5th non-split extension by D4 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.5D20, C40.84D4, Q8.5D20, M4(2).35D10, C8○D4.1D5, (C5×D4).22D4, C4.21(C2×D20), C20.44(C2×D4), (C2×C8).82D10, (C5×Q8).22D4, C4○D4.33D10, C53(D4.5D4), C8.41(C5⋊D4), C40.6C416C2, (C2×Dic20)⋊13C2, (C2×C40).68C22, C4.12D2014C2, C10.78(C4⋊D4), C2.26(C207D4), (C2×C20).423C23, D4.9D10.1C2, C22.10(C4○D20), C4.Dic5.18C22, (C5×M4(2)).38C22, (C2×Dic10).122C22, (C5×C8○D4).1C2, C4.119(C2×C5⋊D4), (C2×C10).8(C4○D4), (C5×C4○D4).38C22, (C2×C4).125(C22×D5), SmallGroup(320,770)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.5D20
C1C5C10C20C2×C20C2×Dic10C2×Dic20 — D4.5D20
C5C10C2×C20 — D4.5D20
C1C2C2×C4C8○D4

Generators and relations for D4.5D20
 G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c19 >

Subgroups: 350 in 100 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4, D4, Q8, Q8 [×4], C10, C10 [×2], C2×C8, C2×C8, M4(2), M4(2) [×3], SD16 [×2], Q16 [×4], C2×Q8 [×2], C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22 [×2], C52C8 [×2], C40 [×2], C40, Dic10 [×4], C2×Dic5 [×2], C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, D4.5D4, Dic20 [×2], C4.Dic5 [×2], D4.D5 [×2], C5⋊Q16 [×2], C2×C40, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10 [×2], C5×C4○D4, C40.6C4, C4.12D20 [×2], C2×Dic20, D4.9D10 [×2], C5×C8○D4, D4.5D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, D4.5D4, C2×D20, C4○D20, C2×C5⋊D4, C207D4, D4.5D20

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

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B8A8B8C8D8E8F8G10A10B10C···10H20A20B20C20D20E···20J40A···40H40I···40T
order122244444558888888101010···102020202020···2040···4040···40
size1124224404022224444040224···422224···42···24···4

56 irreducible representations

dim11111122222222222244
type+++++++++++++++--
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4D20D20C4○D20D4.5D4D4.5D20
kernelD4.5D20C40.6C4C4.12D20C2×Dic20D4.9D10C5×C8○D4C40C5×D4C5×Q8C8○D4C2×C10C2×C8M4(2)C4○D4C8D4Q8C22C5C1
# reps11212121122222844828

Matrix representation of D4.5D20 in GL4(𝔽41) generated by

101517
540390
011822
24161723
,
101517
540390
001822
001723
,
13000
101900
1602638
10036
,
2531335
13355
7351628
35301038
G:=sub<GL(4,GF(41))| [1,5,0,24,0,40,1,16,15,39,18,17,17,0,22,23],[1,5,0,0,0,40,0,0,15,39,18,17,17,0,22,23],[13,10,16,10,0,19,0,0,0,0,26,3,0,0,38,6],[25,13,7,35,31,3,35,30,33,5,16,10,5,5,28,38] >;

D4.5D20 in GAP, Magma, Sage, TeX

D_4._5D_{20}
% in TeX

G:=Group("D4.5D20");
// GroupNames label

G:=SmallGroup(320,770);
// by ID

G=gap.SmallGroup(320,770);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,344,254,1123,297,136,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^20=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=a^2*c^19>;
// generators/relations

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