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G = D203D4order 320 = 26·5

3rd semidirect product of D20 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D203D4, Dic52D8, (C2×D40)⋊5C2, C51(C4⋊D8), C2.11(D5×D8), C4.88(D4×D5), D4⋊C48D5, C20⋊D42C2, C10.25(C2×D8), (C2×C8).12D10, D208C43C2, D206C48C2, C4⋊C4.140D10, (C2×D4).31D10, C4.3(C4○D20), C20.112(C2×D4), C20.8Q87C2, C20.11(C4○D4), (C2×C40).12C22, (C2×Dic5).32D4, C22.183(D4×D5), C10.19(C4⋊D4), C2.13(D40⋊C2), C10.58(C8⋊C22), (C2×C20).226C23, (D4×C10).47C22, (C2×D20).58C22, C2.22(D10⋊D4), (C4×Dic5).19C22, (C2×D4⋊D5)⋊6C2, (C5×D4⋊C4)⋊8C2, (C2×C10).239(C2×D4), (C5×C4⋊C4).27C22, (C2×C52C8).24C22, (C2×C4).333(C22×D5), SmallGroup(320,413)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D203D4
C1C5C10C2×C10C2×C20C2×D20D208C4 — D203D4
C5C10C2×C20 — D203D4
C1C22C2×C4D4⋊C4

Generators and relations for D203D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a8b, dbd=a3b, dcd=c-1 >

Subgroups: 758 in 140 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×11], C23 [×3], D5 [×3], C10 [×3], C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×4], C22×C4, C2×D4, C2×D4 [×4], Dic5 [×2], Dic5, C20 [×2], C20, D10 [×7], C2×C10, C2×C10 [×3], D4⋊C4, D4⋊C4, C4⋊C8, C4×D4, C41D4, C2×D8 [×2], C52C8, C40, C4×D5 [×2], D20 [×2], D20 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C22×D5 [×2], C22×C10, C4⋊D8, D40 [×2], C2×C52C8, C4×Dic5, D10⋊C4, D4⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20 [×2], C2×C5⋊D4 [×2], D4×C10, D206C4, C20.8Q8, C5×D4⋊C4, D208C4, C2×D40, C2×D4⋊D5, C20⋊D4, D203D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C8⋊C22, C22×D5, C4⋊D8, C4○D20, D4×D5 [×2], D10⋊D4, D5×D8, D40⋊C2, D203D4

Smallest permutation representation of D203D4
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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(41 103)(42 102)(43 101)(44 120)(45 119)(46 118)(47 117)(48 116)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 108)(57 107)(58 106)(59 105)(60 104)(61 124)(62 123)(63 122)(64 121)(65 140)(66 139)(67 138)(68 137)(69 136)(70 135)(71 134)(72 133)(73 132)(74 131)(75 130)(76 129)(77 128)(78 127)(79 126)(80 125)(81 143)(82 142)(83 141)(84 160)(85 159)(86 158)(87 157)(88 156)(89 155)(90 154)(91 153)(92 152)(93 151)(94 150)(95 149)(96 148)(97 147)(98 146)(99 145)(100 144)
(1 84 47 125)(2 93 48 134)(3 82 49 123)(4 91 50 132)(5 100 51 121)(6 89 52 130)(7 98 53 139)(8 87 54 128)(9 96 55 137)(10 85 56 126)(11 94 57 135)(12 83 58 124)(13 92 59 133)(14 81 60 122)(15 90 41 131)(16 99 42 140)(17 88 43 129)(18 97 44 138)(19 86 45 127)(20 95 46 136)(21 160 109 80)(22 149 110 69)(23 158 111 78)(24 147 112 67)(25 156 113 76)(26 145 114 65)(27 154 115 74)(28 143 116 63)(29 152 117 72)(30 141 118 61)(31 150 119 70)(32 159 120 79)(33 148 101 68)(34 157 102 77)(35 146 103 66)(36 155 104 75)(37 144 105 64)(38 153 106 73)(39 142 107 62)(40 151 108 71)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 156)(62 155)(63 154)(64 153)(65 152)(66 151)(67 150)(68 149)(69 148)(70 147)(71 146)(72 145)(73 144)(74 143)(75 142)(76 141)(77 160)(78 159)(79 158)(80 157)(81 128)(82 127)(83 126)(84 125)(85 124)(86 123)(87 122)(88 121)(89 140)(90 139)(91 138)(92 137)(93 136)(94 135)(95 134)(96 133)(97 132)(98 131)(99 130)(100 129)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222222444444455888810···1010101010202020202020202040···40
size111182020402244101020224420202···28888444488884···4

47 irreducible representations

dim1111111122222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5D8C4○D4D10D10D10C4○D20C8⋊C22D4×D5D4×D5D5×D8D40⋊C2
kernelD203D4D206C4C20.8Q8C5×D4⋊C4D208C4C2×D40C2×D4⋊D5C20⋊D4D20C2×Dic5D4⋊C4Dic5C20C4⋊C4C2×C8C2×D4C4C10C4C22C2C2
# reps1111111122242222812244

Matrix representation of D203D4 in GL4(𝔽41) generated by

04000
13500
0019
001840
,
283900
21300
002426
001117
,
23600
211800
00400
00040
,
1000
64000
0010
001840
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,1,18,0,0,9,40],[28,2,0,0,39,13,0,0,0,0,24,11,0,0,26,17],[23,21,0,0,6,18,0,0,0,0,40,0,0,0,0,40],[1,6,0,0,0,40,0,0,0,0,1,18,0,0,0,40] >;

D203D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_3D_4
% in TeX

G:=Group("D20:3D4");
// GroupNames label

G:=SmallGroup(320,413);
// by ID

G=gap.SmallGroup(320,413);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,1684,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

׿
×
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