metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D20, C4⋊D5, C5⋊1D4, C20⋊1C2, D10⋊1C2, C2.4D10, C10.3C22, sometimes denoted D40 or Dih20 or Dih40, SmallGroup(40,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20
G = < a,b | a20=b2=1, bab=a-1 >
Character table of D20
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | 10B | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 0 | 0 | -2 | -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 D10 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | -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 |
| ρ8 | 2 | 2 | 0 | 0 | -2 | -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 D10 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | -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 |
| ρ10 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | orthogonal faithful |
| ρ11 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | orthogonal faithful |
| ρ12 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | orthogonal faithful |
| ρ13 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | orthogonal faithful |
►On 20 points - transitive group 20T10
(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);
D20 is a maximal subgroup of
C40⋊C2 D40 D4⋊D5 Q8⋊D5 C4○D20 D4×D5 Q8⋊2D5 C3⋊D20 D60 D100 C5⋊D20 C20⋊D5 C7⋊D20 D140 C32⋊D20 C11⋊D20 D220
D20 is a maximal quotient of
C40⋊C2 D40 Dic20 C4⋊Dic5 D10⋊C4 C3⋊D20 D60 D100 C5⋊D20 C20⋊D5 C7⋊D20 D140 C32⋊D20 C11⋊D20 D220
Matrix representation of D20 ►in GL2(𝔽19) generated by
| 0 | 18 |
| 1 | 11 |
| 11 | 6 |
| 18 | 8 |
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
Export
Subgroup lattice of D20 in TeX
Character table of D20 in TeX