metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10, C5⋊Q8, C4.D5, C20.1C2, C2.3D10, C10.1C22, Dic5.1C2, SmallGroup(40,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10
G = < a,b | a20=1, b2=a10, bab-1=a-1 >
Character table of Dic10
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 10A | 10B | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 2 | 10 | 10 | 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 | 2 | 0 | 0 | -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 |
| ρ6 | 2 | 2 | -2 | 0 | 0 | -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 | -2 | 0 | 0 | -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 |
| ρ8 | 2 | 2 | 2 | 0 | 0 | -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 |
| ρ9 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ10 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | symplectic faithful, Schur index 2 |
| ρ11 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | symplectic faithful, Schur index 2 |
| ρ12 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | symplectic faithful, Schur index 2 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | symplectic faithful, Schur index 2 |
►Regular action on 40 points
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)
(1 23 11 33)(2 22 12 32)(3 21 13 31)(4 40 14 30)(5 39 15 29)(6 38 16 28)(7 37 17 27)(8 36 18 26)(9 35 19 25)(10 34 20 24)
G:=sub<Sym(40)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,23,11,33)(2,22,12,32)(3,21,13,31)(4,40,14,30)(5,39,15,29)(6,38,16,28)(7,37,17,27)(8,36,18,26)(9,35,19,25)(10,34,20,24)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,23,11,33)(2,22,12,32)(3,21,13,31)(4,40,14,30)(5,39,15,29)(6,38,16,28)(7,37,17,27)(8,36,18,26)(9,35,19,25)(10,34,20,24) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)], [(1,23,11,33),(2,22,12,32),(3,21,13,31),(4,40,14,30),(5,39,15,29),(6,38,16,28),(7,37,17,27),(8,36,18,26),(9,35,19,25),(10,34,20,24)]])
Dic10 is a maximal subgroup of
C40⋊C2 Dic20 D4.D5 C5⋊Q16 C4○D20 D4⋊2D5 Q8×D5 C15⋊Q8 Dic30 Dic50 C52⋊2Q8 C52⋊4Q8 C35⋊Q8 Dic70 C32⋊Dic10 C55⋊Q8 Dic110
Dic10 is a maximal quotient of
C10.D4 C4⋊Dic5 C15⋊Q8 Dic30 Dic50 C52⋊2Q8 C52⋊4Q8 C35⋊Q8 Dic70 C32⋊Dic10 C55⋊Q8 Dic110
Matrix representation of Dic10 ►in GL2(𝔽19) generated by
| 8 | 17 |
| 17 | 3 |
| 0 | 18 |
| 1 | 0 |
G:=sub<GL(2,GF(19))| [8,17,17,3],[0,1,18,0] >;
Dic10 in GAP, Magma, Sage, TeX
{\rm Dic}_{10} % in TeX
G:=Group("Dic10"); // GroupNames label
G:=SmallGroup(40,4);
// by ID
G=gap.SmallGroup(40,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-5,16,49,21,515]);
// Polycyclic
G:=Group<a,b|a^20=1,b^2=a^10,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic10 in TeX
Character table of Dic10 in TeX