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G = D4.D20order 320 = 26·5

2nd non-split extension by D4 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.7D20, Dic10.9D4, (C5×D4).2D4, C20.3(C2×D4), C4.87(D4×D5), C4.4(C2×D20), D4⋊C46D5, C4⋊C4.14D10, (C2×C8).10D10, D101C86C2, D102Q84C2, (C2×Dic20)⋊5C2, C10.Q167C2, C52(D4.7D4), C10.22C22≀C2, (C2×D4).137D10, C10.24(C4○D8), (C2×C40).10C22, (C22×D5).23D4, C22.181(D4×D5), C2.10(D83D5), (C2×C20).223C23, (C2×Dic5).199D4, (D4×C10).44C22, C2.25(C22⋊D20), C2.13(SD16⋊D5), C10.31(C8.C22), (C2×Dic10).64C22, (C2×D4.D5)⋊6C2, (C5×D4⋊C4)⋊6C2, (C2×C4×D5).19C22, (C2×D42D5).5C2, (C2×C10).236(C2×D4), (C5×C4⋊C4).24C22, (C2×C52C8).21C22, (C2×C4).330(C22×D5), SmallGroup(320,410)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.D20
C1C5C10C2×C10C2×C20C2×C4×D5C2×D42D5 — D4.D20
C5C10C2×C20 — D4.D20
C1C22C2×C4D4⋊C4

Generators and relations for D4.D20
 G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 638 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×10], D4 [×2], D4 [×5], Q8 [×5], C23 [×2], D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×2], Q16 [×2], C22×C4 [×2], C2×D4, C2×D4, C2×Q8 [×2], C4○D4 [×4], Dic5 [×4], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×4], C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C52C8, C40, Dic10 [×2], Dic10 [×3], C4×D5 [×2], C2×Dic5, C2×Dic5 [×6], C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×C10, D4.7D4, Dic20 [×2], C2×C52C8, C4⋊Dic5, D10⋊C4, D4.D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10 [×2], C2×C4×D5, D42D5 [×4], C22×Dic5, C2×C5⋊D4, D4×C10, C10.Q16, D101C8, C5×D4⋊C4, D102Q8, C2×Dic20, C2×D4.D5, C2×D42D5, D4.D20
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, D4.7D4, C2×D20, D4×D5 [×2], C22⋊D20, D83D5, SD16⋊D5, D4.D20

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444444455888810···1010101010202020202020202040···40
size111144202281010202040224420202···28888444488884···4

47 irreducible representations

dim11111111222222222244444
type+++++++++++++++++-++--
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10D10C4○D8D20C8.C22D4×D5D4×D5D83D5SD16⋊D5
kernelD4.D20C10.Q16D101C8C5×D4⋊C4D102Q8C2×Dic20C2×D4.D5C2×D42D5Dic10C2×Dic5C5×D4C22×D5D4⋊C4C4⋊C4C2×C8C2×D4C10D4C10C4C22C2C2
# reps11111111212122224812244

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

32000
40900
0010
0001
,
192700
142200
00400
00040
,
20900
242100
002730
001132
,
20900
12100
001114
00930
G:=sub<GL(4,GF(41))| [32,40,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[19,14,0,0,27,22,0,0,0,0,40,0,0,0,0,40],[20,24,0,0,9,21,0,0,0,0,27,11,0,0,30,32],[20,1,0,0,9,21,0,0,0,0,11,9,0,0,14,30] >;

D4.D20 in GAP, Magma, Sage, TeX

D_4.D_{20}
% in TeX

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

G:=SmallGroup(320,410);
// by ID

G=gap.SmallGroup(320,410);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,226,851,438,102,12550]);
// Polycyclic

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

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