metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic15, C6.D5, C3⋊Dic5, C15⋊3C4, C10.S3, C2.D15, C5⋊2Dic3, C30.1C2, SmallGroup(60,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — Dic15 |
Generators and relations for Dic15
G = < a,b | a30=1, b2=a15, bab-1=a-1 >
Character table of Dic15
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 6 | 10A | 10B | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 2 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | i | -i | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | -i | i | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 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 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ6 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | ζ3ζ54-ζ3ζ5-ζ5 | ζ32ζ54-ζ32ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | orthogonal lifted from D15 |
| ρ8 | 2 | 2 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | ζ32ζ54-ζ32ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | ζ3ζ54-ζ3ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5-ζ5 | orthogonal lifted from D15 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 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 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | 2 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | ζ3ζ54-ζ3ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ32ζ54-ζ32ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5-ζ5 | orthogonal lifted from D15 |
| ρ11 | 2 | 2 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ32ζ54-ζ32ζ5-ζ5 | ζ3ζ54-ζ3ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | orthogonal lifted from D15 |
| ρ12 | 2 | -2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -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 | 1+√5/2 | 1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ13 | 2 | -2 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | ζ32ζ54-ζ32ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5+ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ3ζ54-ζ3ζ5+ζ54 | symplectic faithful, Schur index 2 |
| ρ14 | 2 | -2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -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 | 1-√5/2 | 1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ15 | 2 | -2 | -1 | 0 | 0 | 2 | 2 | 1 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ16 | 2 | -2 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5-ζ5 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ32ζ54-ζ32ζ5+ζ54 | ζ3ζ54-ζ3ζ5+ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | symplectic faithful, Schur index 2 |
| ρ17 | 2 | -2 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | ζ3ζ54-ζ3ζ5-ζ5 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5+ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ3ζ53-ζ3ζ52+ζ53 | ζ32ζ54-ζ32ζ5+ζ54 | symplectic faithful, Schur index 2 |
| ρ18 | 2 | -2 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5-ζ5 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5-ζ5 | ζ3ζ53-ζ3ζ52+ζ53 | ζ3ζ54-ζ3ζ5+ζ54 | ζ32ζ54-ζ32ζ5+ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | symplectic faithful, Schur index 2 |
►Regular action on 60 points
Generators in S60
(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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)
(1 45 16 60)(2 44 17 59)(3 43 18 58)(4 42 19 57)(5 41 20 56)(6 40 21 55)(7 39 22 54)(8 38 23 53)(9 37 24 52)(10 36 25 51)(11 35 26 50)(12 34 27 49)(13 33 28 48)(14 32 29 47)(15 31 30 46)
G:=sub<Sym(60)| (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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,45,16,60)(2,44,17,59)(3,43,18,58)(4,42,19,57)(5,41,20,56)(6,40,21,55)(7,39,22,54)(8,38,23,53)(9,37,24,52)(10,36,25,51)(11,35,26,50)(12,34,27,49)(13,33,28,48)(14,32,29,47)(15,31,30,46)>;
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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,45,16,60)(2,44,17,59)(3,43,18,58)(4,42,19,57)(5,41,20,56)(6,40,21,55)(7,39,22,54)(8,38,23,53)(9,37,24,52)(10,36,25,51)(11,35,26,50)(12,34,27,49)(13,33,28,48)(14,32,29,47)(15,31,30,46) );
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),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,45,16,60),(2,44,17,59),(3,43,18,58),(4,42,19,57),(5,41,20,56),(6,40,21,55),(7,39,22,54),(8,38,23,53),(9,37,24,52),(10,36,25,51),(11,35,26,50),(12,34,27,49),(13,33,28,48),(14,32,29,47),(15,31,30,46)]])
Dic15 is a maximal subgroup of
D5×Dic3 S3×Dic5 C15⋊D4 C15⋊Q8 Dic30 C4×D15 C15⋊7D4 Dic45 C3⋊Dic15 Q8.D15 A4⋊Dic5 Dic75 C30.D5 D5.D15 Dic105
Dic15 is a maximal quotient of
C15⋊3C8 Dic45 C3⋊Dic15 A4⋊Dic5 Dic75 C30.D5 D5.D15 Dic105
Matrix representation of Dic15 ►in GL2(𝔽29) generated by
| 10 | 7 |
| 7 | 5 |
| 12 | 8 |
| 0 | 17 |
G:=sub<GL(2,GF(29))| [10,7,7,5],[12,0,8,17] >;
Dic15 in GAP, Magma, Sage, TeX
{\rm Dic}_{15} % in TeX
G:=Group("Dic15"); // GroupNames label
G:=SmallGroup(60,3);
// by ID
G=gap.SmallGroup(60,3);
# by ID
G:=PCGroup([4,-2,-2,-3,-5,8,98,771]);
// Polycyclic
G:=Group<a,b|a^30=1,b^2=a^15,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic15 in TeX
Character table of Dic15 in TeX