Copied to
clipboard

## G = D7order 14 = 2·7

### Dihedral group

Aliases: D7, C7⋊C2, sometimes denoted D14 or Dih7 or Dih14, SmallGroup(14,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7
 Chief series C1 — C7 — D7
 Lower central C7 — D7
 Upper central C1

Generators and relations for D7
G = < a,b | a7=b2=1, bab=a-1 >

Character table of D7

 class 1 2 7A 7B 7C size 1 7 2 2 2 ρ1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 linear of order 2 ρ3 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal faithful ρ4 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal faithful ρ5 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal faithful

Permutation representations of D7
On 7 points: primitive - transitive group 7T2
Generators in S7
```(1 2 3 4 5 6 7)
(1 7)(2 6)(3 5)```

`G:=sub<Sym(7)| (1,2,3,4,5,6,7), (1,7)(2,6)(3,5)>;`

`G:=Group( (1,2,3,4,5,6,7), (1,7)(2,6)(3,5) );`

`G=PermutationGroup([(1,2,3,4,5,6,7)], [(1,7),(2,6),(3,5)])`

`G:=TransitiveGroup(7,2);`

Regular action on 14 points - transitive group 14T2
Generators in S14
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)```

`G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)>;`

`G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13) );`

`G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13)])`

`G:=TransitiveGroup(14,2);`

D7 is a maximal subgroup of
F7  C7⋊D7
D7p: D21  D35  D49  D77  D91  D119  D133  D161 ...
D7 is a maximal quotient of
Dic7  C7⋊D7
D7p: D21  D35  D49  D77  D91  D119  D133  D161 ...

Polynomial with Galois group D7 over ℚ
actionf(x)Disc(f)
7T2x7-2x6-x5+x4+x3+x2-x-1-713
14T2x14+9x12+53x10+333x8+1251x6+731x4+5415x2+8591-2182·74·112·717

Matrix representation of D7 in GL2(𝔽13) generated by

 11 8 5 12
,
 12 0 8 1
`G:=sub<GL(2,GF(13))| [11,5,8,12],[12,8,0,1] >;`

D7 in GAP, Magma, Sage, TeX

`D_7`
`% in TeX`

`G:=Group("D7");`
`// GroupNames label`

`G:=SmallGroup(14,1);`
`// by ID`

`G=gap.SmallGroup(14,1);`
`# by ID`

`G:=PCGroup([2,-2,-7,49]);`
`// Polycyclic`

`G:=Group<a,b|a^7=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

Export

׿
×
𝔽