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G = D4.13D20order 320 = 26·5

3rd non-split extension by D4 of D20 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.13D20, Q8.13D20, C20.63C24, C40.12C23, D20.26C23, D40.14C22, M4(2).28D10, Dic20.10C22, Dic10.26C23, C8○D45D5, C51(Q8○D8), (C5×D4).25D4, C4.29(C2×D20), C20.75(C2×D4), (C5×Q8).25D4, C4○D4.40D10, (C2×C8).102D10, D407C213C2, C4.60(C23×D5), C22.5(C2×D20), C8.54(C22×D5), (C2×Dic20)⋊15C2, C8.D1012C2, (C2×C40).70C22, C40⋊C2.2C22, C2.32(C22×D20), C10.30(C22×D4), D4.10D104C2, (C2×C20).517C23, C4○D20.27C22, (C5×M4(2)).30C22, (C2×Dic10).200C22, (C5×C8○D4)⋊5C2, (C2×C10).10(C2×D4), (C5×C4○D4).47C22, (C2×C4).228(C22×D5), SmallGroup(320,1425)

Series: Derived Chief Lower central Upper central

C1C20 — D4.13D20
C1C5C10C20D20C4○D20D4.10D10 — D4.13D20
C5C10C20 — D4.13D20
C1C2C4○D4C8○D4

Generators and relations for D4.13D20
 G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=a2c19 >

Subgroups: 902 in 248 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, C40, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, Q8○D8, C40⋊C2, D40, Dic20, C2×C40, C5×M4(2), C2×Dic10, C4○D20, D42D5, Q8×D5, C5×C4○D4, D407C2, C2×Dic20, C8.D10, C5×C8○D4, D4.10D10, D4.13D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, D20, C22×D5, Q8○D8, C2×D20, C23×D5, C22×D20, D4.13D20

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4J5A5B8A8B8C8D8E10A10B10C···10H20A20B20C20D20E···20J40A···40H40I···40T
order122222244444···45588888101010···102020202020···2040···4040···40
size112222020222220···202222444224···422224···42···24···4

62 irreducible representations

dim1111112222222244
type++++++++++++++--
imageC1C2C2C2C2C2D4D4D5D10D10D10D20D20Q8○D8D4.13D20
kernelD4.13D20D407C2C2×Dic20C8.D10C5×C8○D4D4.10D10C5×D4C5×Q8C8○D4C2×C8M4(2)C4○D4D4Q8C5C1
# reps13361231266212428

Matrix representation of D4.13D20 in GL4(𝔽41) generated by

0010
0001
40000
04000
,
33141525
2781626
1525827
16261433
,
35300
381500
00353
003815
,
203900
152100
002039
001521
G:=sub<GL(4,GF(41))| [0,0,40,0,0,0,0,40,1,0,0,0,0,1,0,0],[33,27,15,16,14,8,25,26,15,16,8,14,25,26,27,33],[35,38,0,0,3,15,0,0,0,0,35,38,0,0,3,15],[20,15,0,0,39,21,0,0,0,0,20,15,0,0,39,21] >;

D4.13D20 in GAP, Magma, Sage, TeX

D_4._{13}D_{20}
% in TeX

G:=Group("D4.13D20");
// GroupNames label

G:=SmallGroup(320,1425);
// by ID

G=gap.SmallGroup(320,1425);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,192,1684,102,12550]);
// Polycyclic

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

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