metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic5, C5⋊2C4, C2.D5, C10.C2, SmallGroup(20,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — Dic5 |
Generators and relations for Dic5
G = < a,b | a10=1, b2=a5, bab-1=a-1 >
Character table of Dic5
| class | 1 | 2 | 4A | 4B | 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 | -i | i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| 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 | symplectic faithful, Schur index 2 |
| ρ8 | 2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | symplectic faithful, Schur index 2 |
►Regular action on 20 points - transitive group 20T2
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 14 6 19)(2 13 7 18)(3 12 8 17)(4 11 9 16)(5 20 10 15)
G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,14,6,19)(2,13,7,18)(3,12,8,17)(4,11,9,16)(5,20,10,15)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,14,6,19)(2,13,7,18)(3,12,8,17)(4,11,9,16)(5,20,10,15) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,14,6,19),(2,13,7,18),(3,12,8,17),(4,11,9,16),(5,20,10,15)]])
G:=TransitiveGroup(20,2);
Dic5 is a maximal subgroup of
C5⋊C8 C4×D5 C5⋊D4 C52⋊6C4 D5.D5 SL2(𝔽5) C32⋊Dic5 C65⋊C4 2- 1+4.D5 C25.D5 C85⋊C4
Dic5p: Dic10 Dic15 Dic25 Dic35 Dic55 Dic65 Dic85 Dic95 ...
Dic5 is a maximal quotient of
D5.D5 C32⋊Dic5 C65⋊C4 C25.D5 C85⋊C4
C2p.D5: C5⋊2C8 Dic15 Dic25 C52⋊6C4 Dic35 Dic55 Dic65 Dic85 ...
Matrix representation of Dic5 ►in GL2(𝔽11) generated by
| 9 | 3 |
| 3 | 6 |
| 0 | 10 |
| 1 | 0 |
G:=sub<GL(2,GF(11))| [9,3,3,6],[0,1,10,0] >;
Dic5 in GAP, Magma, Sage, TeX
{\rm Dic}_5 % in TeX
G:=Group("Dic5"); // GroupNames label
G:=SmallGroup(20,1);
// by ID
G=gap.SmallGroup(20,1);
# by ID
G:=PCGroup([3,-2,-2,-5,6,146]);
// Polycyclic
G:=Group<a,b|a^10=1,b^2=a^5,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic5 in TeX
Character table of Dic5 in TeX