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
| C5 — D10 |
Generators and relations for D10
G = < a,b | a10=b2=1, bab=a-1 >
Character table of D10
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal faithful |
| ρ6 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ7 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ8 | 2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal faithful |
►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);
D10 is a maximal subgroup of
D20 C5⋊D4 2- 1+4⋊D5
D10 is a maximal quotient of Dic10 D20 C5⋊D4
| action | f(x) | Disc(f) |
|---|---|---|
| 10T3 | x10-x9-16x8+11x7+58x6-19x5-68x4+8x3+21x2-3x-1 | 36·55·132·4014 |
Matrix representation of D10 ►in GL2(𝔽11) generated by
| 4 | 1 |
| 10 | 0 |
| 0 | 1 |
| 1 | 0 |
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
Export
Subgroup lattice of D10 in TeX
Character table of D10 in TeX