metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊1D4, D6⋊1D5, D10⋊1S3, C6.4D10, C10.4D6, Dic15⋊4C2, C30.4C22, (C6×D5)⋊1C2, C5⋊2(C3⋊D4), C3⋊2(C5⋊D4), C2.4(S3×D5), (S3×C10)⋊1C2, SmallGroup(120,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊D4
G = < a,b,c | a15=b4=c2=1, bab-1=a-1, cac=a4, cbc=b-1 >
Character table of C15⋊D4
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 15A | 15B | 30A | 30B | |
| size | 1 | 1 | 6 | 10 | 2 | 30 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | -2 | -1 | 0 | 2 | 2 | -1 | 1 | 1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 2 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | -2 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | 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 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ9 | 2 | 2 | 2 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | 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 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | 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 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ11 | 2 | 2 | 2 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | 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 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | -2 | 0 | 0 | -1 | 0 | 2 | 2 | 1 | √-3 | -√-3 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ13 | 2 | -2 | 0 | 0 | -1 | 0 | 2 | 2 | 1 | -√-3 | √-3 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | 0 | 1-√5/2 | 1+√5/2 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | ζ53-ζ52 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | -2 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ17 | 2 | -2 | 0 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | -2 | 0 | 0 | 1+√5/2 | 1-√5/2 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | -ζ54+ζ5 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
| ρ18 | 4 | 4 | 0 | 0 | -2 | 0 | -1-√5 | -1+√5 | -2 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from S3×D5 |
| ρ19 | 4 | 4 | 0 | 0 | -2 | 0 | -1+√5 | -1-√5 | -2 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from S3×D5 |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | symplectic faithful, Schur index 2 |
►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 38 20 59)(2 37 21 58)(3 36 22 57)(4 35 23 56)(5 34 24 55)(6 33 25 54)(7 32 26 53)(8 31 27 52)(9 45 28 51)(10 44 29 50)(11 43 30 49)(12 42 16 48)(13 41 17 47)(14 40 18 46)(15 39 19 60)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)(16 19)(17 23)(18 27)(21 24)(22 28)(26 29)(31 46)(32 50)(33 54)(34 58)(35 47)(36 51)(37 55)(38 59)(39 48)(40 52)(41 56)(42 60)(43 49)(44 53)(45 57)
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), (1,38,20,59)(2,37,21,58)(3,36,22,57)(4,35,23,56)(5,34,24,55)(6,33,25,54)(7,32,26,53)(8,31,27,52)(9,45,28,51)(10,44,29,50)(11,43,30,49)(12,42,16,48)(13,41,17,47)(14,40,18,46)(15,39,19,60), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)(16,19)(17,23)(18,27)(21,24)(22,28)(26,29)(31,46)(32,50)(33,54)(34,58)(35,47)(36,51)(37,55)(38,59)(39,48)(40,52)(41,56)(42,60)(43,49)(44,53)(45,57)>;
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), (1,38,20,59)(2,37,21,58)(3,36,22,57)(4,35,23,56)(5,34,24,55)(6,33,25,54)(7,32,26,53)(8,31,27,52)(9,45,28,51)(10,44,29,50)(11,43,30,49)(12,42,16,48)(13,41,17,47)(14,40,18,46)(15,39,19,60), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)(16,19)(17,23)(18,27)(21,24)(22,28)(26,29)(31,46)(32,50)(33,54)(34,58)(35,47)(36,51)(37,55)(38,59)(39,48)(40,52)(41,56)(42,60)(43,49)(44,53)(45,57) );
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)], [(1,38,20,59),(2,37,21,58),(3,36,22,57),(4,35,23,56),(5,34,24,55),(6,33,25,54),(7,32,26,53),(8,31,27,52),(9,45,28,51),(10,44,29,50),(11,43,30,49),(12,42,16,48),(13,41,17,47),(14,40,18,46),(15,39,19,60)], [(2,5),(3,9),(4,13),(7,10),(8,14),(12,15),(16,19),(17,23),(18,27),(21,24),(22,28),(26,29),(31,46),(32,50),(33,54),(34,58),(35,47),(36,51),(37,55),(38,59),(39,48),(40,52),(41,56),(42,60),(43,49),(44,53),(45,57)]])
C15⋊D4 is a maximal subgroup of
D20⋊5S3 D6.D10 D12⋊5D5 C20⋊D6 C30.C23 D5×C3⋊D4 S3×C5⋊D4 C45⋊D4 C30.12D6 D6⋊D15 D30⋊S3 D10.1S4 D10⋊S4
C15⋊D4 is a maximal quotient of
C15⋊D8 C30.D4 C20.D6 C15⋊Q16 D10⋊Dic3 D6⋊Dic5 Dic15⋊5C4 C45⋊D4 C30.12D6 D6⋊D15 D30⋊S3 D10⋊S4
Matrix representation of C15⋊D4 ►in GL4(𝔽61) generated by
| 60 | 17 | 0 | 0 |
| 44 | 44 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 2 | 13 |
| 14 | 45 | 0 | 0 |
| 39 | 47 | 0 | 0 |
| 0 | 0 | 55 | 41 |
| 0 | 0 | 11 | 6 |
| 1 | 0 | 0 | 0 |
| 17 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 36 | 60 |
G:=sub<GL(4,GF(61))| [60,44,0,0,17,44,0,0,0,0,47,2,0,0,0,13],[14,39,0,0,45,47,0,0,0,0,55,11,0,0,41,6],[1,17,0,0,0,60,0,0,0,0,1,36,0,0,0,60] >;
C15⋊D4 in GAP, Magma, Sage, TeX
C_{15}\rtimes D_4 % in TeX
G:=Group("C15:D4"); // GroupNames label
G:=SmallGroup(120,11);
// by ID
G=gap.SmallGroup(120,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,61,168,2404]);
// Polycyclic
G:=Group<a,b,c|a^15=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^4,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C15⋊D4 in TeX
Character table of C15⋊D4 in TeX