Copied to
clipboard

G = D2014D4order 320 = 26·5

2nd semidirect product of D20 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2014D4, Dic1013D4, C23.12D20, (C2×D40)⋊3C2, C22⋊C86D5, (C2×C8).3D10, C51(D4⋊D4), C4.122(D4×D5), D205C49C2, C207D416C2, C10.9(C4○D8), (C2×C20).241D4, (C2×C4).119D20, C20.334(C2×D4), C10.10C22≀C2, C20.44D46C2, (C22×C10).54D4, (C22×C4).84D10, C2.13(C8⋊D10), C10.10(C8⋊C22), (C2×C20).744C23, (C2×C40).119C22, C22.107(C2×D20), C4⋊Dic5.13C22, C2.13(C22⋊D20), C2.11(D407C2), (C2×D20).199C22, (C22×C20).97C22, (C2×Dic10).217C22, (C2×C4○D20)⋊1C2, (C5×C22⋊C8)⋊8C2, (C2×C40⋊C2)⋊11C2, (C2×C10).127(C2×D4), (C2×C4).689(C22×D5), SmallGroup(320,361)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D2014D4
C1C5C10C20C2×C20C2×D20C2×C4○D20 — D2014D4
C5C10C2×C20 — D2014D4
C1C22C22×C4C22⋊C8

Generators and relations for D2014D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a13b, dbd=a10b, dcd=c-1 >

Subgroups: 830 in 162 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4 [×2], C2×C4 [×8], D4 [×11], Q8 [×3], C23, C23 [×2], D5 [×3], C10 [×3], C10, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C22×C4, C2×D4 [×4], C2×Q8, C4○D4 [×4], Dic5 [×3], C20 [×2], C20, D10 [×7], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C40 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, D4⋊D4, C40⋊C2 [×2], D40 [×2], C4⋊Dic5, D10⋊C4, C2×C40 [×2], C2×Dic10, C2×C4×D5, C2×D20 [×2], C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C20.44D4, D205C4, C5×C22⋊C8, C2×C40⋊C2, C2×D40, C207D4, C2×C4○D20, D2014D4
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⋊D4, C2×D20, D4×D5 [×2], C22⋊D20, D407C2, C8⋊D10, D2014D4

Smallest permutation representation of D2014D4
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 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 121)(22 140)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 112)(58 111)(59 110)(60 109)(61 160)(62 159)(63 158)(64 157)(65 156)(66 155)(67 154)(68 153)(69 152)(70 151)(71 150)(72 149)(73 148)(74 147)(75 146)(76 145)(77 144)(78 143)(79 142)(80 141)
(1 77 103 30)(2 76 104 29)(3 75 105 28)(4 74 106 27)(5 73 107 26)(6 72 108 25)(7 71 109 24)(8 70 110 23)(9 69 111 22)(10 68 112 21)(11 67 113 40)(12 66 114 39)(13 65 115 38)(14 64 116 37)(15 63 117 36)(16 62 118 35)(17 61 119 34)(18 80 120 33)(19 79 101 32)(20 78 102 31)(41 124 87 156)(42 123 88 155)(43 122 89 154)(44 121 90 153)(45 140 91 152)(46 139 92 151)(47 138 93 150)(48 137 94 149)(49 136 95 148)(50 135 96 147)(51 134 97 146)(52 133 98 145)(53 132 99 144)(54 131 100 143)(55 130 81 142)(56 129 82 141)(57 128 83 160)(58 127 84 159)(59 126 85 158)(60 125 86 157)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(41 149)(42 150)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 157)(50 158)(51 159)(52 160)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222222444444455888810···101010101020···202020202040···40
size1111420204022222020402244442···244442···244444···4

59 irreducible representations

dim1111111122222222222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10C4○D8D20D20D407C2C8⋊C22D4×D5C8⋊D10
kernelD2014D4C20.44D4D205C4C5×C22⋊C8C2×C40⋊C2C2×D40C207D4C2×C4○D20Dic10D20C2×C20C22×C10C22⋊C8C2×C8C22×C4C10C2×C4C23C2C10C4C2
# reps11111111221124244416144

Matrix representation of D2014D4 in GL6(𝔽41)

010000
4000000
0004000
0013500
000010
000001
,
29290000
29120000
00283900
0021300
000010
000001
,
090000
900000
0040000
0035100
0000224
00001219
,
090000
3200000
001000
000100
0000224
00003319

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[29,29,0,0,0,0,29,12,0,0,0,0,0,0,28,2,0,0,0,0,39,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,22,12,0,0,0,0,4,19],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,33,0,0,0,0,4,19] >;

D2014D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{14}D_4
% in TeX

G:=Group("D20:14D4");
// GroupNames label

G:=SmallGroup(320,361);
// by ID

G=gap.SmallGroup(320,361);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,226,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^13*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽