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G = D80:7C2  order 320 = 26·5

The semidirect product of D80 and C2 acting through Inn(D80)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D80:7C2, C8.13D20, C4.20D40, C40.63D4, C20.38D8, Dic40:7C2, C16.16D10, C22.1D40, C40.57C23, C80.18C22, D40.7C22, Dic20.7C22, (C2xC16):6D5, (C2xC80):10C2, C5:1(C4oD16), C16:D5:7C2, C4.38(C2xD20), (C2xC10).20D8, C10.11(C2xD8), C2.13(C2xD40), (C2xC4).85D20, D40:7C2:1C2, (C2xC20).395D4, (C2xC8).313D10, C20.281(C2xD4), C8.47(C22xD5), (C2xC40).385C22, SmallGroup(320,531)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C40 — D80:7C2
Chief series
C1 — C5 — C10 — C20 — C40 — D40 — D40:7C2 — D80:7C2
Lower central
C5 — C10 — C20 — C40 — D80:7C2
Upper central
C1 — C4 — C2xC4 — C2xC8 — C2xC16

Generators and relations for D80:7C2
 G = < a,b,c | a80=b2=c2=1, bab=a-1, ac=ca, cbc=a40b >

Subgroups: 478 in 84 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, Q8, D5, C10, C10, C16, C2xC8, D8, SD16, Q16, C4oD4, Dic5, C20, D10, C2xC10, C2xC16, D16, SD32, Q32, C4oD8, C40, Dic10, C4xD5, D20, C5:D4, C2xC20, C4oD16, C80, C40:C2, D40, Dic20, C2xC40, C4oD20, D80, C16:D5, Dic40, C2xC80, D40:7C2, D80:7C2
Quotients: C1, C2, C22, D4, C23, D5, D8, C2xD4, D10, C2xD8, D20, C22xD5, C4oD16, D40, C2xD20, C2xD40, D80:7C2

Smallest permutation representation of D80:7C2
►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 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(81 85)(82 84)(86 160)(87 159)(88 158)(89 157)(90 156)(91 155)(92 154)(93 153)(94 152)(95 151)(96 150)(97 149)(98 148)(99 147)(100 146)(101 145)(102 144)(103 143)(104 142)(105 141)(106 140)(107 139)(108 138)(109 137)(110 136)(111 135)(112 134)(113 133)(114 132)(115 131)(116 130)(117 129)(118 128)(119 127)(120 126)(121 125)(122 124)

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

86 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B8A8B8C8D10A···10F16A···16H20A···20H40A···40P80A···80AF
order122224444455888810···1016···1620···2040···4080···80
size112404011240402222222···22···22···22···22···2

86 irreducible representations

dim1111112222222222222
type+++++++++++++++++
imageC1C2C2C2C2C2D4D4D5D8D8D10D10D20D20C4oD16D40D40D80:7C2
kernelD80:7C2D80C16:D5Dic40C2xC80D40:7C2C40C2xC20C2xC16C20C2xC10C16C2xC8C8C2xC4C5C4C22C1
# reps11211211222424488832

Matrix representation of D80:7C2 ►in GL2(F241) generated by

175173
6881
,
569
214185
,
7649
192165
G:=sub<GL(2,GF(241))| [175,68,173,81],[56,214,9,185],[76,192,49,165] >;
 

D80:7C2 in GAP, Magma, Sage, TeX 

D_{80}\rtimes_7C_2
 
% in TeX
 
G:=Group("D80:7C2");
 
// GroupNames label
 
G:=SmallGroup(320,531);
 
// by ID
 
G=gap.SmallGroup(320,531);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,142,675,192,1684,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c|a^80=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^40*b>;
 
// generators/relations
 

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