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G = D807C2order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D807C2, C8.13D20, C4.20D40, C40.63D4, C20.38D8, Dic407C2, C16.16D10, C22.1D40, C40.57C23, C80.18C22, D40.7C22, Dic20.7C22, (C2×C16)⋊6D5, (C2×C80)⋊10C2, C51(C4○D16), C16⋊D57C2, C4.38(C2×D20), (C2×C10).20D8, C10.11(C2×D8), C2.13(C2×D40), (C2×C4).85D20, D407C21C2, (C2×C20).395D4, (C2×C8).313D10, C20.281(C2×D4), C8.47(C22×D5), (C2×C40).385C22, SmallGroup(320,531)

Series: Derived Chief Lower central Upper central

C1C40 — D807C2
C1C5C10C20C40D40D407C2 — D807C2
C5C10C20C40 — D807C2
C1C4C2×C4C2×C8C2×C16

Generators and relations for D807C2
 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 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D5 [×2], C10, C10, C16 [×2], C2×C8, D8 [×2], SD16 [×2], Q16 [×2], C4○D4 [×2], Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C2×C16, D16, SD32 [×2], Q32, C4○D8 [×2], C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, C4○D16, C80 [×2], C40⋊C2 [×2], D40 [×2], Dic20 [×2], C2×C40, C4○D20 [×2], D80, C16⋊D5 [×2], Dic40, C2×C80, D407C2 [×2], D807C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, D20 [×2], C22×D5, C4○D16, D40 [×2], C2×D20, C2×D40, D807C2

Smallest permutation representation of D807C2
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)(81 145)(82 144)(83 143)(84 142)(85 141)(86 140)(87 139)(88 138)(89 137)(90 136)(91 135)(92 134)(93 133)(94 132)(95 131)(96 130)(97 129)(98 128)(99 127)(100 126)(101 125)(102 124)(103 123)(104 122)(105 121)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(146 160)(147 159)(148 158)(149 157)(150 156)(151 155)(152 154)
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 141)(57 142)(58 143)(59 144)(60 145)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 81)(77 82)(78 83)(79 84)(80 85)

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

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

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

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+++++++++++++++++
imageC1C2C2C2C2C2D4D4D5D8D8D10D10D20D20C4○D16D40D40D807C2
kernelD807C2D80C16⋊D5Dic40C2×C80D407C2C40C2×C20C2×C16C20C2×C10C16C2×C8C8C2×C4C5C4C22C1
# reps11211211222424488832

Matrix representation of D807C2 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] >;

D807C2 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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