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G = D2024D4order 320 = 26·5

2nd semidirect product of D20 and D4 acting through Inn(D20)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2024D4, C42.109D10, C10.592- 1+4, (C4×D4)⋊13D5, (D4×C20)⋊15C2, (C4×D20)⋊29C2, C202(C4○D4), C43(C4○D20), C202D48C2, C52(D46D4), C4.140(D4×D5), C4⋊C4.316D10, C202Q824C2, D10.38(C2×D4), C20.346(C2×D4), (C2×D4).214D10, (C2×C10).95C24, C10.50(C22×D4), D10.12D46C2, C20.48D420C2, (C2×C20).783C23, (C4×C20).152C22, C22⋊C4.110D10, (C22×C4).208D10, C23.95(C22×D5), (C2×D20).295C22, (D4×C10).257C22, C4⋊Dic5.199C22, (C2×Dic5).41C23, C22.120(C23×D5), C23.D5.12C22, D10⋊C4.98C22, (C22×C20).107C22, (C22×C10).165C23, C10.D4.65C22, (C22×D5).183C23, C2.16(D4.10D10), (C2×Dic10).247C22, C2.23(C2×D4×D5), (D5×C4⋊C4)⋊15C2, (C2×C4○D20)⋊8C2, C10.42(C2×C4○D4), C2.46(C2×C4○D20), (C2×C4×D5).73C22, (C5×C4⋊C4).326C22, (C2×C4).579(C22×D5), (C2×C5⋊D4).121C22, (C5×C22⋊C4).122C22, SmallGroup(320,1223)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D2024D4
C1C5C10C2×C10C22×D5C2×C4×D5D5×C4⋊C4 — D2024D4
C5C2×C10 — D2024D4
C1C22C4×D4

Generators and relations for D2024D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 1030 in 292 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×14], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×14], Q8 [×4], C23 [×2], C23 [×2], D5 [×4], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8 [×2], C4○D4 [×8], Dic5 [×6], C20 [×4], C20 [×3], D10 [×4], D10 [×4], C2×C10, C2×C10 [×6], C2×C4⋊C4 [×2], C4×D4, C4×D4, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×12], D20 [×4], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], D46D4, C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×6], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C202Q8, C4×D20, D10.12D4 [×4], D5×C4⋊C4 [×2], C20.48D4 [×2], C202D4 [×2], D4×C20, C2×C4○D20 [×2], D2024D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, C22×D5 [×7], D46D4, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D4.10D10, D2024D4

Smallest permutation representation of D2024D4
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 43)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 80)(77 79)(81 89)(82 88)(83 87)(84 86)(90 100)(91 99)(92 98)(93 97)(94 96)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(121 125)(122 124)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)
(1 98 61 143)(2 99 62 144)(3 100 63 145)(4 81 64 146)(5 82 65 147)(6 83 66 148)(7 84 67 149)(8 85 68 150)(9 86 69 151)(10 87 70 152)(11 88 71 153)(12 89 72 154)(13 90 73 155)(14 91 74 156)(15 92 75 157)(16 93 76 158)(17 94 77 159)(18 95 78 160)(19 96 79 141)(20 97 80 142)(21 131 60 118)(22 132 41 119)(23 133 42 120)(24 134 43 101)(25 135 44 102)(26 136 45 103)(27 137 46 104)(28 138 47 105)(29 139 48 106)(30 140 49 107)(31 121 50 108)(32 122 51 109)(33 123 52 110)(34 124 53 111)(35 125 54 112)(36 126 55 113)(37 127 56 114)(38 128 57 115)(39 129 58 116)(40 130 59 117)
(1 118)(2 119)(3 120)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 143)(22 144)(23 145)(24 146)(25 147)(26 148)(27 149)(28 150)(29 151)(30 152)(31 153)(32 154)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 141)(40 142)(41 99)(42 100)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I4J···4O5A5B10A···10F10G···10N20A···20H20I···20X
order12222222224···444···45510···1010···1020···2020···20
size111144101010102···2420···20222···24···42···24···4

65 irreducible representations

dim111111111222222222444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D10D10C4○D202- 1+4D4×D5D4.10D10
kernelD2024D4C202Q8C4×D20D10.12D4D5×C4⋊C4C20.48D4C202D4D4×C20C2×C4○D20D20C4×D4C20C42C22⋊C4C4⋊C4C22×C4C2×D4C4C10C4C2
# reps1114222124242424216144

Matrix representation of D2024D4 in GL6(𝔽41)

3200000
3790000
00344000
008100
000010
000001
,
40250000
010000
00404000
000100
000010
000001
,
100000
010000
0040000
0004000
0000040
000010
,
100000
5400000
0040000
0004000
000010
0000040

G:=sub<GL(6,GF(41))| [32,37,0,0,0,0,0,9,0,0,0,0,0,0,34,8,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,25,1,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,5,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

D2024D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{24}D_4
% in TeX

G:=Group("D20:24D4");
// GroupNames label

G:=SmallGroup(320,1223);
// by ID

G=gap.SmallGroup(320,1223);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,675,570,12550]);
// Polycyclic

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

׿
×
𝔽