direct product, metacyclic, supersoluble, monomial
Aliases: C3×D12, C12⋊1C6, C12⋊3S3, D6⋊1C6, C32⋊4D4, C6.19D6, C4⋊(C3×S3), C3⋊1(C3×D4), (S3×C6)⋊3C2, (C3×C12)⋊2C2, C2.4(S3×C6), C6.3(C2×C6), (C3×C6).8C22, SmallGroup(72,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D12
G = < a,b,c | a3=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C3×D12
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 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 | 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 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ6 | ζ32 | ζ65 | ζ65 | ζ65 | -1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ65 | ζ3 | ζ6 | ζ6 | ζ6 | -1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ6 | ζ3 | ζ65 | ζ32 | ζ6 | ζ6 | -1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ8 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ10 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ11 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ65 | ζ32 | ζ6 | ζ3 | ζ65 | ζ65 | -1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | √3 | -√3 | √3 | -√3 | 0 | √3 | -√3 | 0 | orthogonal lifted from D12 |
| ρ17 | 2 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -√3 | √3 | -√3 | √3 | 0 | -√3 | √3 | 0 | orthogonal lifted from D12 |
| ρ18 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | -2 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | ζ32 | ζ32 | 1 | 1 | 1-√-3 | ζ3 | ζ3 | 1+√-3 | complex lifted from S3×C6 |
| ρ19 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 2 | 0 | 1+√-3 | 1-√-3 | 1-√-3 | 1+√-3 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ20 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 2 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | -1 | -1 | -1+√-3 | ζ65 | ζ65 | -1-√-3 | complex lifted from C3×S3 |
| ρ21 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | -2 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | ζ3 | ζ3 | 1 | 1 | 1+√-3 | ζ32 | ζ32 | 1-√-3 | complex lifted from S3×C6 |
| ρ22 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 1+√-3 | 1-√-3 | ζ3 | ζ32 | 1 | 0 | 0 | 0 | 0 | ζ4ζ32+2ζ4 | ζ43ζ32+2ζ43 | -√3 | √3 | 0 | ζ43ζ3+2ζ43 | ζ4ζ3+2ζ4 | 0 | complex faithful |
| ρ23 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 1-√-3 | 1+√-3 | ζ32 | ζ3 | 1 | 0 | 0 | 0 | 0 | ζ4ζ3+2ζ4 | ζ43ζ3+2ζ43 | √3 | -√3 | 0 | ζ43ζ32+2ζ43 | ζ4ζ32+2ζ4 | 0 | complex faithful |
| ρ24 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 2 | 0 | 1-√-3 | 1+√-3 | 1+√-3 | 1-√-3 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ25 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 2 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | -1 | -1 | -1-√-3 | ζ6 | ζ6 | -1+√-3 | complex lifted from C3×S3 |
| ρ26 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 1-√-3 | 1+√-3 | ζ32 | ζ3 | 1 | 0 | 0 | 0 | 0 | ζ43ζ3+2ζ43 | ζ4ζ3+2ζ4 | -√3 | √3 | 0 | ζ4ζ32+2ζ4 | ζ43ζ32+2ζ43 | 0 | complex faithful |
| ρ27 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 1+√-3 | 1-√-3 | ζ3 | ζ32 | 1 | 0 | 0 | 0 | 0 | ζ43ζ32+2ζ43 | ζ4ζ32+2ζ4 | √3 | -√3 | 0 | ζ4ζ3+2ζ4 | ζ43ζ3+2ζ43 | 0 | complex faithful |
►On 24 points - transitive group 24T67
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 24)(9 23)(10 22)(11 21)(12 20)
G:=sub<Sym(24)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,24)(9,23)(10,22)(11,21)(12,20)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,24)(9,23)(10,22)(11,21)(12,20) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,24),(9,23),(10,22),(11,21),(12,20)]])
G:=TransitiveGroup(24,67);
C3×D12 is a maximal subgroup of
C32⋊2D8 C3⋊D24 Dic6⋊S3 D12.S3 D12⋊5S3 D12⋊S3 D6⋊D6 C3×S3×D4 He3⋊4D4 D36⋊C3 He3⋊5D4 D12.A4
C3×D12 is a maximal quotient of
He3⋊4D4 D36⋊C3
Matrix representation of C3×D12 ►in GL2(𝔽13) generated by
| 9 | 0 |
| 0 | 9 |
| 6 | 0 |
| 4 | 11 |
| 2 | 9 |
| 4 | 11 |
G:=sub<GL(2,GF(13))| [9,0,0,9],[6,4,0,11],[2,4,9,11] >;
C3×D12 in GAP, Magma, Sage, TeX
C_3\times D_{12} % in TeX
G:=Group("C3xD12"); // GroupNames label
G:=SmallGroup(72,28);
// by ID
G=gap.SmallGroup(72,28);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-3,141,66,1204]);
// Polycyclic
G:=Group<a,b,c|a^3=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×D12 in TeX
Character table of C3×D12 in TeX