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G = D₅  order 10 = 2·5

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D₅, C₅⋊C₂, sometimes denoted D₁₀ or Dih₅ or Dih₁₀, symmetries of a regular pentagon, SmallGroup(10,1)

Series: Derived ►Chief ►Lower central ►Upper central

Generators and relations for D₅
 G = < a,b | a⁵=b²=1, bab=a^-1 >

Character table of D₅

(1 2 3 4 5)

(1 5)(2 4)

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

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

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

G:=TransitiveGroup(5,2);

(1 2 3 4 5)(6 7 8 9 10)

(1 8)(2 7)(3 6)(4 10)(5 9)

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

G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,8)(2,7)(3,6)(4,10)(5,9) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,8),(2,7),(3,6),(4,10),(5,9)]])

G:=TransitiveGroup(10,2);

D₅ is a maximal subgroup of
 F₅  C₅⋊D₅  A₅  C₂⁴⋊D₅
 D_5p: D₁₅  D₂₅  D₃₅  D₅₅  D₆₅  D₈₅  D₉₅  D₁₁₅ ...
D₅ is a maximal quotient of
 Dic₅  C₅⋊D₅  C₂⁴⋊D₅
 D_5p: D₁₅  D₂₅  D₃₅  D₅₅  D₆₅  D₈₅  D₉₅  D₁₁₅ ...

Matrix representation of D₅ ►in GL₂(𝔽₁₁) generated by

G:=sub<GL(2,GF(11))| [3,1,10,0],[3,8,10,8] >;

D₅ in GAP, Magma, Sage, TeX

D_5

% in TeX

G:=Group("D5");

// GroupNames label

G:=SmallGroup(10,1);

// by ID

G=gap.SmallGroup(10,1);

# by ID

G:=PCGroup([2,-2,-5,33]);

// Polycyclic

G:=Group<a,b|a^5=b^2=1,b*a*b=a^-1>;

// generators/relations

Export

Subgroup lattice of D₅ in TeX
Character table of D₅ in TeX