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G = D₄  order 8 = 2³

Dihedral group

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

Aliases: D₄, He₂, 2₊¹⁺², C₂≀C₂, 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

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

Character table of D₄

(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);

(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
 D_4p: D₈  D₁₂  D₂₀  D₂₈  D₄₄  D₅₂  D₆₈  D₇₆ ...
 D_2p⋊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
 D_4p: D₈  D₁₂  D₂₀  D₂₈  D₄₄  D₅₂  D₆₈  D₇₆ ...
 C₂.D_2p: C₂²⋊C₄  C₄⋊C₄  SD₁₆  Q₁₆  C₃⋊D₄  C₅⋊D₄  C₇⋊D₄  C₁₁⋊D₄ ...

Matrix representation of D₄ ►in GL₂(ℤ) generated by

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

Export

Subgroup lattice of D₄ in TeX
Character table of D₄ in TeX