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## G = D80order 160 = 25·5

### Dihedral group

Aliases: D80, C51D16, C801C2, C161D5, D401C2, C4.1D20, C2.3D40, C10.1D8, C20.24D4, C8.13D10, C40.14C22, sometimes denoted D160 or Dih80 or Dih160, SmallGroup(160,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D80
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — D80
 Lower central C5 — C10 — C20 — C40 — D80
 Upper central C1 — C2 — C4 — C8 — C16

Generators and relations for D80
G = < a,b | a80=b2=1, bab=a-1 >

40C2
40C2
20C22
20C22
8D5
8D5
10D4
10D4
4D10
4D10
5D8
5D8
2D20
2D20
5D16

Smallest permutation representation of D80
On 80 points
Generators in S80
```(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)
(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)```

`G:=sub<Sym(80)| (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), (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)>;`

`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), (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) );`

`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)], [(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)])`

D80 is a maximal subgroup of
D160  C160⋊C2  C5⋊D32  C5⋊SD64  D807C2  D80⋊C2  D5×D16  C16⋊D10  D805C2  C3⋊D80  D240
D80 is a maximal quotient of
D160  C160⋊C2  Dic80  C8013C4  D407C4  C3⋊D80  D240

43 conjugacy classes

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

43 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 D4 D5 D8 D10 D16 D20 D40 D80 kernel D80 C80 D40 C20 C16 C10 C8 C5 C4 C2 C1 # reps 1 1 2 1 2 2 2 4 4 8 16

Matrix representation of D80 in GL2(𝔽241) generated by

 239 173 230 228
,
 183 102 45 58
`G:=sub<GL(2,GF(241))| [239,230,173,228],[183,45,102,58] >;`

D80 in GAP, Magma, Sage, TeX

`D_{80}`
`% in TeX`

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

`G:=SmallGroup(160,6);`
`// by ID`

`G=gap.SmallGroup(160,6);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,73,79,218,122,579,69,4613]);`
`// Polycyclic`

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

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