direct product, metacyclic, supersoluble, monomial
Aliases: C3×Dic6, C12.1C6, C32⋊3Q8, C12.7S3, C6.17D6, Dic3.C6, C3⋊(C3×Q8), C4.(C3×S3), C2.3(S3×C6), C6.1(C2×C6), (C3×C12).2C2, (C3×C6).6C22, (C3×Dic3).2C2, SmallGroup(72,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic6
G = < a,b,c | a3=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C3×Dic6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 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 | -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 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | -1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | -1 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | -1 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | -1 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | -1 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ8 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | -1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ9 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | -1 | 1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | -1 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | -1 | ζ65 | ζ6 | ζ65 | ζ3 | ζ32 | linear of order 6 |
| ρ10 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ11 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ12 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | -1 | 1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | -1 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | -1 | ζ6 | ζ65 | ζ6 | ζ32 | ζ3 | linear of order 6 |
| ρ13 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ15 | 2 | -2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ16 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | √3 | √3 | -√3 | 0 | √3 | -√3 | -√3 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | -√3 | -√3 | √3 | 0 | -√3 | √3 | √3 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | -2 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | -1-√-3 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 1-√-3 | -2 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
| ρ19 | 2 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | -1 | ζ65 | ζ65 | -1-√-3 | ζ6 | ζ6 | -1 | -1+√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ20 | 2 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 1 | ζ3 | ζ3 | 1+√-3 | ζ32 | ζ32 | 1 | 1-√-3 | 0 | 0 | 0 | 0 | complex lifted from S3×C6 |
| ρ21 | 2 | -2 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | -1+√-3 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 1+√-3 | -2 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
| ρ22 | 2 | -2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | ζ32 | 1 | ζ3 | -√3 | ζ4ζ32+2ζ4 | ζ43ζ32+2ζ43 | 0 | ζ43ζ3+2ζ43 | ζ4ζ3+2ζ4 | √3 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 2 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | -1 | ζ6 | ζ6 | -1+√-3 | ζ65 | ζ65 | -1 | -1-√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ24 | 2 | -2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | ζ32 | 1 | ζ3 | √3 | ζ43ζ32+2ζ43 | ζ4ζ32+2ζ4 | 0 | ζ4ζ3+2ζ4 | ζ43ζ3+2ζ43 | -√3 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 2 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 1 | ζ32 | ζ32 | 1-√-3 | ζ3 | ζ3 | 1 | 1+√-3 | 0 | 0 | 0 | 0 | complex lifted from S3×C6 |
| ρ26 | 2 | -2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | ζ3 | 1 | ζ32 | √3 | ζ4ζ3+2ζ4 | ζ43ζ3+2ζ43 | 0 | ζ43ζ32+2ζ43 | ζ4ζ32+2ζ4 | -√3 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 2 | -2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | ζ3 | 1 | ζ32 | -√3 | ζ43ζ3+2ζ43 | ζ4ζ3+2ζ4 | 0 | ζ4ζ32+2ζ4 | ζ43ζ32+2ζ43 | √3 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T64
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)
G:=sub<Sym(24)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,16,7,22),(2,15,8,21),(3,14,9,20),(4,13,10,19),(5,24,11,18),(6,23,12,17)]])
G:=TransitiveGroup(24,64);
C3×Dic6 is a maximal subgroup of
Dic6⋊S3 C32⋊5SD16 C32⋊2Q16 C32⋊3Q16 D12⋊S3 Dic3.D6 D6.6D6 C3×S3×Q8 He3⋊3Q8 C36.C6 He3⋊4Q8 Dic6.A4
C3×Dic6 is a maximal quotient of
He3⋊3Q8 C36.C6
Matrix representation of C3×Dic6 ►in GL2(𝔽13) generated by
| 3 | 0 |
| 0 | 3 |
| 7 | 0 |
| 0 | 2 |
| 0 | 1 |
| 12 | 0 |
G:=sub<GL(2,GF(13))| [3,0,0,3],[7,0,0,2],[0,12,1,0] >;
C3×Dic6 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_6 % in TeX
G:=Group("C3xDic6"); // GroupNames label
G:=SmallGroup(72,26);
// by ID
G=gap.SmallGroup(72,26);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-3,60,141,66,1204]);
// Polycyclic
G:=Group<a,b,c|a^3=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×Dic6 in TeX
Character table of C3×Dic6 in TeX