direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×Dic3, C3⋊C12, C6.C6, C6.4S3, C32⋊2C4, C2.(C3×S3), (C3×C6).1C2, SmallGroup(36,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C3×Dic3 |
Generators and relations for C3×Dic3
G = < a,b,c | a3=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C3×Dic3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | |
| 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 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | i | -i | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | -i | i | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | linear of order 12 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | -i | i | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | i | -i | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | linear of order 12 |
| ρ13 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 1+√-3 | 1-√-3 | ζ3 | 1 | ζ32 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 2 | -2 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 1-√-3 | 1+√-3 | ζ32 | 1 | ζ3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ17 | 2 | 2 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ18 | 2 | 2 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
►On 12 points - transitive group 12T19
Generators in S12
(1 3 5)(2 4 6)(7 11 9)(8 12 10)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 10 4 7)(2 9 5 12)(3 8 6 11)
G:=sub<Sym(12)| (1,3,5)(2,4,6)(7,11,9)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10,4,7)(2,9,5,12)(3,8,6,11)>;
G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10,4,7)(2,9,5,12)(3,8,6,11) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,10,4,7),(2,9,5,12),(3,8,6,11)]])
G:=TransitiveGroup(12,19);
C3×Dic3 is a maximal subgroup of
C6.D6 C3⋊D12 C32⋊2Q8 S3×C12 C32⋊C12 C9⋊C12 He3⋊3C4 Dic3.A4 C6.F7 He3⋊4Dic3 C39⋊3C12 C3⋊F13
C3×Dic3 is a maximal quotient of
C32⋊C12 C9⋊C12 C6.F7 C39⋊3C12 C3⋊F13
| action | f(x) | Disc(f) |
|---|---|---|
| 12T19 | x12-4x11-15x10+68x9+62x8-362x7-107x6+798x5+112x4-654x3-63x2+84x+1 | 216·59·194·5412·1019992 |
Matrix representation of C3×Dic3 ►in GL2(𝔽7) generated by
| 2 | 0 |
| 0 | 2 |
| 5 | 0 |
| 0 | 3 |
| 0 | 6 |
| 1 | 0 |
G:=sub<GL(2,GF(7))| [2,0,0,2],[5,0,0,3],[0,1,6,0] >;
C3×Dic3 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_3 % in TeX
G:=Group("C3xDic3"); // GroupNames label
G:=SmallGroup(36,6);
// by ID
G=gap.SmallGroup(36,6);
# by ID
G:=PCGroup([4,-2,-3,-2,-3,24,387]);
// Polycyclic
G:=Group<a,b,c|a^3=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×Dic3 in TeX
Character table of C3×Dic3 in TeX