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G = D5order 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

C1C5 — D5
C1C5 — D5
C5 — D5
C1

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

5C2

Character table of D5

 class 125A5B
 size 1522
ρ11111    trivial
ρ21-111    linear of order 2
ρ320-1-5/2-1+5/2    orthogonal faithful
ρ420-1+5/2-1-5/2    orthogonal faithful

Permutation representations of D5
On 5 points: primitive - transitive group 5T2
Generators in S5
(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);

Regular action on 10 points - transitive group 10T2
Generators in S10
(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);

Polynomial with Galois group D5 over ℚ
actionf(x)Disc(f)
5T2x5-5x2-334·56
10T2x10-2x9-20x8+2x7+69x6-x5-69x4+2x3+20x2-2x-1316·4015

Matrix representation of D5 in GL2(𝔽11) generated by

310
10
,
310
88
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

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