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G = D20order 40 = 23·5

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D20, C4⋊D5, C51D4, C201C2, D101C2, C2.4D10, C10.3C22, sometimes denoted D40 or Dih20 or Dih40, SmallGroup(40,6)

Series: Derived Chief Lower central Upper central

C1C10 — D20
C1C5C10D10 — D20
C5C10 — D20
C1C2C4

Generators and relations for D20
 G = < a,b | a20=b2=1, bab=a-1 >

10C2
10C2
5C22
5C22
2D5
2D5
5D4

Character table of D20

 class 12A2B2C45A5B10A10B20A20B20C20D
 size 111010222222222
ρ11111111111111    trivial
ρ211-11-11111-1-1-1-1    linear of order 2
ρ3111-1-11111-1-1-1-1    linear of order 2
ρ411-1-1111111111    linear of order 2
ρ52-200022-2-20000    orthogonal lifted from D4
ρ62200-2-1+5/2-1-5/2-1+5/2-1-5/21+5/21-5/21-5/21+5/2    orthogonal lifted from D10
ρ722002-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ82200-2-1-5/2-1+5/2-1-5/2-1+5/21-5/21+5/21+5/21-5/2    orthogonal lifted from D10
ρ922002-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ102-2000-1-5/2-1+5/21+5/21-5/243ζ5443ζ54ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ5    orthogonal faithful
ρ112-2000-1+5/2-1-5/21-5/21+5/2ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ54ζ534ζ52    orthogonal faithful
ρ122-2000-1+5/2-1-5/21-5/21+5/24ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5ζ4ζ534ζ52    orthogonal faithful
ρ132-2000-1-5/2-1+5/21+5/21-5/2ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ5243ζ5443ζ5    orthogonal faithful

Permutation representations of D20
On 20 points - transitive group 20T10
Generators in S20
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)

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

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

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

G:=TransitiveGroup(20,10);

Matrix representation of D20 in GL2(𝔽19) generated by

018
111
,
116
188
G:=sub<GL(2,GF(19))| [0,1,18,11],[11,18,6,8] >;

D20 in GAP, Magma, Sage, TeX

D_{20}
% in TeX

G:=Group("D20");
// GroupNames label

G:=SmallGroup(40,6);
// by ID

G=gap.SmallGroup(40,6);
# by ID

G:=PCGroup([4,-2,-2,-2,-5,49,21,515]);
// Polycyclic

G:=Group<a,b|a^20=b^2=1,b*a*b=a^-1>;
// generators/relations

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