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## G = D4order 8 = 23

### Dihedral group

p-group, metacyclic, nilpotent (class 2), monomial, rational

Aliases: D4, He2, 2+ 1+2, C2C2, AΣL1(𝔽4), C4⋊C2, C22⋊C2, C2.1C22, 2-Sylow(S4), sometimes denoted D8 or Dih4 or Dih8, symmetries of a square, SmallGroup(8,3)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — D4
 Chief series C1 — C2 — C22 — D4
 Lower central C1 — C2 — D4
 Upper central C1 — C2 — D4
 Jennings C1 — C2 — D4

Generators and relations for D4
G = < a,b | a4=b2=1, bab=a-1 >

Character table of D4

 class 1 2A 2B 2C 4 size 1 1 2 2 2 ρ1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 linear of order 2 ρ3 1 1 1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 linear of order 2 ρ5 2 -2 0 0 0 orthogonal faithful

Permutation representations of D4
On 4 points - transitive group 4T3
Generators in S4
```(1 2 3 4)
(1 4)(2 3)```

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

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

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

`G:=TransitiveGroup(4,3);`

Regular action on 8 points - transitive group 8T4
Generators in S8
```(1 2 3 4)(5 6 7 8)
(1 8)(2 7)(3 6)(4 5)```

`G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5)>;`

`G:=Group( (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5) );`

`G=PermutationGroup([[(1,2,3,4),(5,6,7,8)], [(1,8),(2,7),(3,6),(4,5)]])`

`G:=TransitiveGroup(8,4);`

D4 is a maximal subgroup of
SD16  S4  S3≀C2  D5≀C2  D7≀C2
D4p: D8  D12  D20  D28  D44  D52  D68  D76 ...
D2p⋊C2: C4○D4  C3⋊D4  C5⋊D4  C7⋊D4  C11⋊D4  C13⋊D4  C17⋊D4  C19⋊D4 ...
D4 is a maximal quotient of
S3≀C2  D5≀C2  D7≀C2  C4⋊S5  C22⋊S5
D4p: D8  D12  D20  D28  D44  D52  D68  D76 ...
C2.D2p: C22⋊C4  C4⋊C4  SD16  Q16  C3⋊D4  C5⋊D4  C7⋊D4  C11⋊D4 ...

Polynomial with Galois group D4 over ℚ
actionf(x)Disc(f)
4T3x4-2-211
8T4x8-3x7-11x6+27x5+38x4-51x3-29x2+11x+1212·54·72·132·294

Matrix representation of D4 in GL2(ℤ) generated by

 0 -1 1 0
,
 1 0 0 -1
`G:=sub<GL(2,Integers())| [0,1,-1,0],[1,0,0,-1] >;`

D4 in GAP, Magma, Sage, TeX

`D_4`
`% in TeX`

`G:=Group("D4");`
`// GroupNames label`

`G:=SmallGroup(8,3);`
`// by ID`

`G=gap.SmallGroup(8,3);`
`# by ID`

`G:=PCGroup([3,-2,2,-2,37]);`
`// Polycyclic`

`G:=Group<a,b|a^4=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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