metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2D12, C15⋊3D4, D6⋊2D5, Dic5⋊S3, D30⋊4C2, C10.6D6, C6.6D10, C30.6C22, C3⋊1(C5⋊D4), C2.6(S3×D5), (S3×C10)⋊2C2, (C3×Dic5)⋊3C2, SmallGroup(120,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊D12
G = < a,b,c | a5=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C5⋊D12
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 30A | 30B | |
| size | 1 | 1 | 6 | 30 | 2 | 10 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 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 | 0 | 2 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 0 | 0 | -1 | -2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | -2 | 0 | 0 | -1 | 0 | 2 | 2 | 1 | -2 | -2 | 0 | 0 | 0 | 0 | -√3 | √3 | -1 | -1 | 1 | 1 | orthogonal lifted from D12 |
| ρ9 | 2 | 2 | -2 | 0 | 2 | 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 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ10 | 2 | 2 | 2 | 0 | 2 | 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 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ11 | 2 | 2 | 2 | 0 | 2 | 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 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 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 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | -2 | 0 | 0 | -1 | 0 | 2 | 2 | 1 | -2 | -2 | 0 | 0 | 0 | 0 | √3 | -√3 | -1 | -1 | 1 | 1 | orthogonal lifted from D12 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | -2 | 1+√5/2 | 1-√5/2 | ζ54-ζ5 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | 0 | 0 | -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 | 1-√5/2 | 1+√5/2 | -ζ53+ζ52 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | 0 | 0 | -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 | 1+√5/2 | 1-√5/2 | -ζ54+ζ5 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | 0 | 0 | -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 | 1-√5/2 | 1+√5/2 | ζ53-ζ52 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | 0 | 0 | -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 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal faithful, Schur index 2 |
| ρ19 | 4 | 4 | 0 | 0 | -2 | 0 | -1-√5 | -1+√5 | -2 | -1-√5 | -1+√5 | 0 | 0 | 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 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal faithful, Schur index 2 |
| ρ21 | 4 | 4 | 0 | 0 | -2 | 0 | -1+√5 | -1-√5 | -2 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from S3×D5 |
►On 60 points
Generators in S60
(1 28 50 20 46)(2 47 21 51 29)(3 30 52 22 48)(4 37 23 53 31)(5 32 54 24 38)(6 39 13 55 33)(7 34 56 14 40)(8 41 15 57 35)(9 36 58 16 42)(10 43 17 59 25)(11 26 60 18 44)(12 45 19 49 27)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 53)(14 52)(15 51)(16 50)(17 49)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)
G:=sub<Sym(60)| (1,28,50,20,46)(2,47,21,51,29)(3,30,52,22,48)(4,37,23,53,31)(5,32,54,24,38)(6,39,13,55,33)(7,34,56,14,40)(8,41,15,57,35)(9,36,58,16,42)(10,43,17,59,25)(11,26,60,18,44)(12,45,19,49,27), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,53)(14,52)(15,51)(16,50)(17,49)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46)>;
G:=Group( (1,28,50,20,46)(2,47,21,51,29)(3,30,52,22,48)(4,37,23,53,31)(5,32,54,24,38)(6,39,13,55,33)(7,34,56,14,40)(8,41,15,57,35)(9,36,58,16,42)(10,43,17,59,25)(11,26,60,18,44)(12,45,19,49,27), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,53)(14,52)(15,51)(16,50)(17,49)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46) );
G=PermutationGroup([[(1,28,50,20,46),(2,47,21,51,29),(3,30,52,22,48),(4,37,23,53,31),(5,32,54,24,38),(6,39,13,55,33),(7,34,56,14,40),(8,41,15,57,35),(9,36,58,16,42),(10,43,17,59,25),(11,26,60,18,44),(12,45,19,49,27)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,53),(14,52),(15,51),(16,50),(17,49),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,48),(35,47),(36,46)]])
C5⋊D12 is a maximal subgroup of
D12⋊D5 D60⋊C2 D6.D10 D5×D12 Dic3.D10 S3×C5⋊D4 D10⋊D6 C5⋊D36 C15⋊D12 D6⋊2D15 D30⋊S3 GL2(𝔽3)⋊D5 Dic5⋊S4
C5⋊D12 is a maximal quotient of
C5⋊D24 D12.D5 Dic6⋊D5 C5⋊Dic12 D6⋊Dic5 D30⋊4C4 C30.Q8 C5⋊D36 C15⋊D12 D6⋊2D15 D30⋊S3 Dic5⋊S4
Matrix representation of C5⋊D12 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 1 | 0 | 0 |
| 0 | 0 | 39 | 39 |
| 0 | 0 | 47 | 22 |
| 1 | 60 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 17 | 44 |
| 0 | 0 | 60 | 44 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,17,60,0,0,1,0],[0,60,0,0,1,1,0,0,0,0,39,47,0,0,39,22],[1,0,0,0,60,60,0,0,0,0,17,60,0,0,44,44] >;
C5⋊D12 in GAP, Magma, Sage, TeX
C_5\rtimes D_{12} % in TeX
G:=Group("C5:D12"); // GroupNames label
G:=SmallGroup(120,13);
// by ID
G=gap.SmallGroup(120,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,20,61,168,2404]);
// Polycyclic
G:=Group<a,b,c|a^5=b^12=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C5⋊D12 in TeX
Character table of C5⋊D12 in TeX