metabelian, supersoluble, monomial, A-group
Aliases: C6.2D6, Dic3○Dic3, Dic3⋊2S3, C2.2S32, C3⋊S3⋊1C4, C3⋊1(C4×S3), C32⋊3(C2×C4), (C3×Dic3)⋊3C2, (C3×C6).2C22, (C2×C3⋊S3).1C2, SmallGroup(72,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C6.D6 |
Generators and relations for C6.D6
G = < a,b,c | a6=c2=1, b6=a3, bab-1=cac=a-1, cbc=b5 >
Character table of C6.D6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 3 | 3 | 3 | 3 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | |
| 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 | -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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | -i | i | i | -1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | i | -i | -i | -1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 0 | -2 | -2 | 0 | 2 | -1 | -1 | 1 | 0 | 1 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -2 | 0 | 0 | -2 | -1 | 2 | -1 | 0 | 1 | 0 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | -1 | 0 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 0 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 0 | -1 | 0 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | 2i | 0 | 0 | -2i | 1 | -2 | 1 | 0 | -i | 0 | i | complex lifted from C4×S3 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -2i | 0 | 0 | 2i | 1 | -2 | 1 | 0 | i | 0 | -i | complex lifted from C4×S3 |
| ρ15 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 2i | -2i | 0 | -2 | 1 | 1 | i | 0 | -i | 0 | complex lifted from C4×S3 |
| ρ16 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | -2i | 2i | 0 | -2 | 1 | 1 | -i | 0 | i | 0 | complex lifted from C4×S3 |
| ρ17 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ18 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 12 points - transitive group 12T39
(1 3 5 7 9 11)(2 12 10 8 6 4)
(1 2 3 4 5 6 7 8 9 10 11 12)
(1 9)(3 7)(4 12)(6 10)
G:=sub<Sym(12)| (1,3,5,7,9,11)(2,12,10,8,6,4), (1,2,3,4,5,6,7,8,9,10,11,12), (1,9)(3,7)(4,12)(6,10)>;
G:=Group( (1,3,5,7,9,11)(2,12,10,8,6,4), (1,2,3,4,5,6,7,8,9,10,11,12), (1,9)(3,7)(4,12)(6,10) );
G=PermutationGroup([[(1,3,5,7,9,11),(2,12,10,8,6,4)], [(1,2,3,4,5,6,7,8,9,10,11,12)], [(1,9),(3,7),(4,12),(6,10)]])
G:=TransitiveGroup(12,39);
►On 24 points - transitive group 24T75
(1 3 5 7 9 11)(2 12 10 8 6 4)(13 23 21 19 17 15)(14 16 18 20 22 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 24)(2 17)(3 22)(4 15)(5 20)(6 13)(7 18)(8 23)(9 16)(10 21)(11 14)(12 19)
G:=sub<Sym(24)| (1,3,5,7,9,11)(2,12,10,8,6,4)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,24)(2,17)(3,22)(4,15)(5,20)(6,13)(7,18)(8,23)(9,16)(10,21)(11,14)(12,19)>;
G:=Group( (1,3,5,7,9,11)(2,12,10,8,6,4)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,24)(2,17)(3,22)(4,15)(5,20)(6,13)(7,18)(8,23)(9,16)(10,21)(11,14)(12,19) );
G=PermutationGroup([[(1,3,5,7,9,11),(2,12,10,8,6,4),(13,23,21,19,17,15),(14,16,18,20,22,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,24),(2,17),(3,22),(4,15),(5,20),(6,13),(7,18),(8,23),(9,16),(10,21),(11,14),(12,19)]])
G:=TransitiveGroup(24,75);
C6.D6 is a maximal subgroup of
S32⋊C4 C3⋊S3.Q8 Dic3.D6 D6.6D6 C4×S32 D6.3D6 Dic3⋊D6 C18.D6 C6.S32 C33⋊8(C2×C4) C33⋊9(C2×C4) Dic3.5S4 Dic3⋊2S4 C6.D30 Dic15⋊S3 C3⋊F5⋊S3
C6.D6 is a maximal quotient of
C12.29D6 C12.31D6 Dic32 C6.D12 C62.C22 C18.D6 He3⋊(C2×C4) C33⋊8(C2×C4) C33⋊9(C2×C4) Dic3⋊2S4 C6.D30 Dic15⋊S3 C3⋊F5⋊S3
| action | f(x) | Disc(f) |
|---|---|---|
| 12T39 | x12-8x10+24x8-32x6+17x4-8x2+8 | 241·74·234 |
Matrix representation of C6.D6 ►in GL4(ℤ) generated by
| 1 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | -1 | -1 |
| 0 | 0 | 0 | 1 |
| 0 | -1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | -1 |
G:=sub<GL(4,Integers())| [1,-1,0,0,1,0,0,0,0,0,1,-1,0,0,1,0],[0,0,0,-1,0,0,-1,0,-1,0,0,0,-1,1,0,0],[0,-1,0,0,-1,0,0,0,0,0,1,0,0,0,1,-1] >;
C6.D6 in GAP, Magma, Sage, TeX
C_6.D_6
% in TeX
G:=Group("C6.D6"); // GroupNames label
G:=SmallGroup(72,21);
// by ID
G=gap.SmallGroup(72,21);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,20,26,168,1204]);
// Polycyclic
G:=Group<a,b,c|a^6=c^2=1,b^6=a^3,b*a*b^-1=c*a*c=a^-1,c*b*c=b^5>;
// generators/relations
Export
Subgroup lattice of C6.D6 in TeX
Character table of C6.D6 in TeX