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## G = D80⋊5C2order 320 = 26·5

### 5th semidirect product of D80 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D80⋊5C2
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — Q8.D10 — D80⋊5C2
 Lower central C5 — C10 — C20 — C40 — D80⋊5C2
 Upper central C1 — C2 — C4 — C8 — Q32

Generators and relations for D805C2
G = < a,b,c | a80=b2=c2=1, bab=a-1, cac=a49, cbc=a8b >

Subgroups: 454 in 84 conjugacy classes, 31 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C16, C16, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, D10, C2×C16, D16, SD32, Q32, C4○D8, C52C8, C40, C4×D5, C4×D5, D20, C5×Q8, C4○D16, C52C16, C80, C8×D5, D40, Q8⋊D5, C5×Q16, Q82D5, D5×C16, D80, C5⋊SD32, C5×Q32, Q8.D10, D805C2
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, C4○D16, D4×D5, D5×D8, D805C2

Smallest permutation representation of D805C2
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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 160)(98 159)(99 158)(100 157)(101 156)(102 155)(103 154)(104 153)(105 152)(106 151)(107 150)(108 149)(109 148)(110 147)(111 146)(112 145)(113 144)(114 143)(115 142)(116 141)(117 140)(118 139)(119 138)(120 137)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)
(1 109)(2 158)(3 127)(4 96)(5 145)(6 114)(7 83)(8 132)(9 101)(10 150)(11 119)(12 88)(13 137)(14 106)(15 155)(16 124)(17 93)(18 142)(19 111)(20 160)(21 129)(22 98)(23 147)(24 116)(25 85)(26 134)(27 103)(28 152)(29 121)(30 90)(31 139)(32 108)(33 157)(34 126)(35 95)(36 144)(37 113)(38 82)(39 131)(40 100)(41 149)(42 118)(43 87)(44 136)(45 105)(46 154)(47 123)(48 92)(49 141)(50 110)(51 159)(52 128)(53 97)(54 146)(55 115)(56 84)(57 133)(58 102)(59 151)(60 120)(61 89)(62 138)(63 107)(64 156)(65 125)(66 94)(67 143)(68 112)(69 81)(70 130)(71 99)(72 148)(73 117)(74 86)(75 135)(76 104)(77 153)(78 122)(79 91)(80 140)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A 10B 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D 80A ··· 80H order 1 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 10 10 16 16 16 16 16 16 16 16 20 20 20 20 20 20 40 40 40 40 80 ··· 80 size 1 1 10 40 40 2 5 5 8 8 2 2 2 2 10 10 2 2 2 2 2 2 10 10 10 10 4 4 16 16 16 16 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 C4○D16 D4×D5 D5×D8 D80⋊5C2 kernel D80⋊5C2 D5×C16 D80 C5⋊SD32 C5×Q32 Q8.D10 C5⋊2C8 C4×D5 Q32 Dic5 D10 C16 Q16 C5 C4 C2 C1 # reps 1 1 1 2 1 2 1 1 2 2 2 2 4 8 2 4 8

Matrix representation of D805C2 in GL4(𝔽241) generated by

 189 240 0 0 1 0 0 0 0 0 183 71 0 0 85 112
,
 189 240 0 0 52 52 0 0 0 0 183 71 0 0 214 58
,
 1 0 0 0 189 240 0 0 0 0 177 113 0 0 64 64
`G:=sub<GL(4,GF(241))| [189,1,0,0,240,0,0,0,0,0,183,85,0,0,71,112],[189,52,0,0,240,52,0,0,0,0,183,214,0,0,71,58],[1,189,0,0,0,240,0,0,0,0,177,64,0,0,113,64] >;`

D805C2 in GAP, Magma, Sage, TeX

`D_{80}\rtimes_5C_2`
`% in TeX`

`G:=Group("D80:5C2");`
`// GroupNames label`

`G:=SmallGroup(320,546);`
`// by ID`

`G=gap.SmallGroup(320,546);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,758,135,184,346,185,192,851,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^80=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^49,c*b*c=a^8*b>;`
`// generators/relations`

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