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## G = D40order 80 = 24·5

### Dihedral group

Aliases: D40, C51D8, C81D5, C401C2, D201C2, C4.9D10, C2.4D20, C10.2D4, C20.9C22, sometimes denoted D80 or Dih40 or Dih80, SmallGroup(80,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D40
 Chief series C1 — C5 — C10 — C20 — D20 — D40
 Lower central C5 — C10 — C20 — D40
 Upper central C1 — C2 — C4 — C8

Generators and relations for D40
G = < a,b | a40=b2=1, bab=a-1 >

20C2
20C2
10C22
10C22
4D5
4D5
5D4
5D4
2D10
2D10
5D8

Character table of D40

 class 1 2A 2B 2C 4 5A 5B 8A 8B 10A 10B 20A 20B 20C 20D 40A 40B 40C 40D 40E 40F 40G 40H size 1 1 20 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 0 -2 2 2 0 0 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 0 0 2 -1+√5/2 -1-√5/2 -2 -2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ7 2 2 0 0 2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ8 2 2 0 0 2 -1-√5/2 -1+√5/2 -2 -2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ9 2 2 0 0 2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ10 2 -2 0 0 0 2 2 -√2 √2 -2 -2 0 0 0 0 √2 √2 -√2 -√2 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ11 2 -2 0 0 0 2 2 √2 -√2 -2 -2 0 0 0 0 -√2 -√2 √2 √2 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ12 2 2 0 0 -2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 orthogonal lifted from D20 ρ13 2 2 0 0 -2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 orthogonal lifted from D20 ρ14 2 2 0 0 -2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 orthogonal lifted from D20 ρ15 2 2 0 0 -2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 orthogonal lifted from D20 ρ16 2 -2 0 0 0 -1-√5/2 -1+√5/2 -√2 √2 1+√5/2 1-√5/2 -ζ82ζ53+ζ82ζ52 -ζ86ζ54+ζ86ζ5 ζ86ζ54-ζ86ζ5 ζ82ζ53-ζ82ζ52 -ζ83ζ53+ζ8ζ52 -ζ83ζ52+ζ8ζ53 ζ83ζ52-ζ8ζ53 -ζ87ζ54+ζ85ζ5 ζ83ζ5-ζ8ζ54 ζ83ζ53-ζ8ζ52 ζ87ζ54-ζ85ζ5 -ζ83ζ5+ζ8ζ54 orthogonal faithful ρ17 2 -2 0 0 0 -1+√5/2 -1-√5/2 -√2 √2 1-√5/2 1+√5/2 ζ86ζ54-ζ86ζ5 -ζ82ζ53+ζ82ζ52 ζ82ζ53-ζ82ζ52 -ζ86ζ54+ζ86ζ5 ζ87ζ54-ζ85ζ5 -ζ83ζ5+ζ8ζ54 ζ83ζ5-ζ8ζ54 ζ83ζ52-ζ8ζ53 ζ83ζ53-ζ8ζ52 -ζ87ζ54+ζ85ζ5 -ζ83ζ52+ζ8ζ53 -ζ83ζ53+ζ8ζ52 orthogonal faithful ρ18 2 -2 0 0 0 -1-√5/2 -1+√5/2 √2 -√2 1+√5/2 1-√5/2 ζ82ζ53-ζ82ζ52 ζ86ζ54-ζ86ζ5 -ζ86ζ54+ζ86ζ5 -ζ82ζ53+ζ82ζ52 ζ83ζ52-ζ8ζ53 ζ83ζ53-ζ8ζ52 -ζ83ζ53+ζ8ζ52 -ζ83ζ5+ζ8ζ54 ζ87ζ54-ζ85ζ5 -ζ83ζ52+ζ8ζ53 ζ83ζ5-ζ8ζ54 -ζ87ζ54+ζ85ζ5 orthogonal faithful ρ19 2 -2 0 0 0 -1+√5/2 -1-√5/2 √2 -√2 1-√5/2 1+√5/2 ζ86ζ54-ζ86ζ5 -ζ82ζ53+ζ82ζ52 ζ82ζ53-ζ82ζ52 -ζ86ζ54+ζ86ζ5 -ζ87ζ54+ζ85ζ5 ζ83ζ5-ζ8ζ54 -ζ83ζ5+ζ8ζ54 -ζ83ζ52+ζ8ζ53 -ζ83ζ53+ζ8ζ52 ζ87ζ54-ζ85ζ5 ζ83ζ52-ζ8ζ53 ζ83ζ53-ζ8ζ52 orthogonal faithful ρ20 2 -2 0 0 0 -1+√5/2 -1-√5/2 √2 -√2 1-√5/2 1+√5/2 -ζ86ζ54+ζ86ζ5 ζ82ζ53-ζ82ζ52 -ζ82ζ53+ζ82ζ52 ζ86ζ54-ζ86ζ5 ζ83ζ5-ζ8ζ54 -ζ87ζ54+ζ85ζ5 ζ87ζ54-ζ85ζ5 -ζ83ζ53+ζ8ζ52 -ζ83ζ52+ζ8ζ53 -ζ83ζ5+ζ8ζ54 ζ83ζ53-ζ8ζ52 ζ83ζ52-ζ8ζ53 orthogonal faithful ρ21 2 -2 0 0 0 -1+√5/2 -1-√5/2 -√2 √2 1-√5/2 1+√5/2 -ζ86ζ54+ζ86ζ5 ζ82ζ53-ζ82ζ52 -ζ82ζ53+ζ82ζ52 ζ86ζ54-ζ86ζ5 -ζ83ζ5+ζ8ζ54 ζ87ζ54-ζ85ζ5 -ζ87ζ54+ζ85ζ5 ζ83ζ53-ζ8ζ52 ζ83ζ52-ζ8ζ53 ζ83ζ5-ζ8ζ54 -ζ83ζ53+ζ8ζ52 -ζ83ζ52+ζ8ζ53 orthogonal faithful ρ22 2 -2 0 0 0 -1-√5/2 -1+√5/2 -√2 √2 1+√5/2 1-√5/2 ζ82ζ53-ζ82ζ52 ζ86ζ54-ζ86ζ5 -ζ86ζ54+ζ86ζ5 -ζ82ζ53+ζ82ζ52 -ζ83ζ52+ζ8ζ53 -ζ83ζ53+ζ8ζ52 ζ83ζ53-ζ8ζ52 ζ83ζ5-ζ8ζ54 -ζ87ζ54+ζ85ζ5 ζ83ζ52-ζ8ζ53 -ζ83ζ5+ζ8ζ54 ζ87ζ54-ζ85ζ5 orthogonal faithful ρ23 2 -2 0 0 0 -1-√5/2 -1+√5/2 √2 -√2 1+√5/2 1-√5/2 -ζ82ζ53+ζ82ζ52 -ζ86ζ54+ζ86ζ5 ζ86ζ54-ζ86ζ5 ζ82ζ53-ζ82ζ52 ζ83ζ53-ζ8ζ52 ζ83ζ52-ζ8ζ53 -ζ83ζ52+ζ8ζ53 ζ87ζ54-ζ85ζ5 -ζ83ζ5+ζ8ζ54 -ζ83ζ53+ζ8ζ52 -ζ87ζ54+ζ85ζ5 ζ83ζ5-ζ8ζ54 orthogonal faithful

Smallest permutation representation of D40
On 40 points
Generators in S40
```(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)
(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)```

`G:=sub<Sym(40)| (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), (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)>;`

`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), (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) );`

`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)], [(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)])`

D40 is a maximal subgroup of
D80  C16⋊D5  C5⋊D16  C5⋊SD32  D407C2  C8⋊D10  D5×D8  D40⋊C2  Q8.D10  C3⋊D40  D120  D200  C5⋊D40  C525D8
D40 is a maximal quotient of
D80  C16⋊D5  Dic40  C405C4  D205C4  C3⋊D40  D120  D200  C5⋊D40  C525D8

Matrix representation of D40 in GL2(𝔽41) generated by

 5 3 38 23
,
 12 18 8 29
`G:=sub<GL(2,GF(41))| [5,38,3,23],[12,8,18,29] >;`

D40 in GAP, Magma, Sage, TeX

`D_{40}`
`% in TeX`

`G:=Group("D40");`
`// GroupNames label`

`G:=SmallGroup(80,7);`
`// by ID`

`G=gap.SmallGroup(80,7);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,61,66,182,42,1604]);`
`// Polycyclic`

`G:=Group<a,b|a^40=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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