Copied to
clipboard

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

C1C2 — D4
C1C2C22 — D4
C1C2 — D4
C1C2 — D4
C1C2 — D4

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

2C2
2C2

Character table of D4

 class 12A2B2C4
 size 11222
ρ111111    trivial
ρ211-11-1    linear of order 2
ρ3111-1-1    linear of order 2
ρ411-1-11    linear of order 2
ρ52-2000    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);

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
10
,
10
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

׿
×
𝔽