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G = D10order 20 = 22·5

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D10, C2×D5, C10⋊C2, C5⋊C22, sometimes denoted D20 or Dih10 or Dih20, SmallGroup(20,4)

Series: Derived Chief Lower central Upper central

C1C5 — D10
C1C5D5 — D10
C5 — D10
C1C2

Generators and relations for D10
 G = < a,b | a10=b2=1, bab=a-1 >

5C2
5C2
5C22

Character table of D10

 class 12A2B2C5A5B10A10B
 size 11552222
ρ111111111    trivial
ρ211-1-11111    linear of order 2
ρ31-1-1111-1-1    linear of order 2
ρ41-11-111-1-1    linear of order 2
ρ52-200-1+5/2-1-5/21+5/21-5/2    orthogonal faithful
ρ62200-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ72200-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ82-200-1-5/2-1+5/21-5/21+5/2    orthogonal faithful

Permutation representations of D10
On 10 points - transitive group 10T3
Generators in S10
(1 2 3 4 5 6 7 8 9 10)
(1 5)(2 4)(6 10)(7 9)

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

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

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

G:=TransitiveGroup(10,3);

Regular action on 20 points - transitive group 20T4
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)

G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,12),(2,11),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13)])

G:=TransitiveGroup(20,4);

Polynomial with Galois group D10 over ℚ
actionf(x)Disc(f)
10T3x10-x9-16x8+11x7+58x6-19x5-68x4+8x3+21x2-3x-136·55·132·4014

Matrix representation of D10 in GL2(𝔽11) generated by

41
100
,
01
10
G:=sub<GL(2,GF(11))| [4,10,1,0],[0,1,1,0] >;

D10 in GAP, Magma, Sage, TeX

D_{10}
% in TeX

G:=Group("D10");
// GroupNames label

G:=SmallGroup(20,4);
// by ID

G=gap.SmallGroup(20,4);
# by ID

G:=PCGroup([3,-2,-2,-5,146]);
// Polycyclic

G:=Group<a,b|a^10=b^2=1,b*a*b=a^-1>;
// generators/relations

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