metabelian, supersoluble, monomial
Aliases: D6⋊2S3, C3⋊2D12, Dic3⋊S3, C6.4D6, C32⋊3D4, C2.4S32, (S3×C6)⋊2C2, C3⋊1(C3⋊D4), (C3×Dic3)⋊1C2, (C3×C6).4C22, (C2×C3⋊S3)⋊1C2, SmallGroup(72,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D12
G = < a,b,c | a3=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C3⋊D12
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4 | 6A | 6B | 6C | 6D | 6E | 12A | 12B | |
| size | 1 | 1 | 6 | 18 | 2 | 2 | 4 | 6 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | 0 | -1 | 2 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | 2 | -1 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | -2 | 1 | 1 | 0 | 0 | √3 | -√3 | orthogonal lifted from D12 |
| ρ11 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | -2 | 1 | 1 | 0 | 0 | -√3 | √3 | orthogonal lifted from D12 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 1 | -2 | 1 | -√-3 | √-3 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 1 | -2 | 1 | √-3 | -√-3 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ14 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ15 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 12 points - transitive group 12T38
(1 9 5)(2 6 10)(3 11 7)(4 8 12)
(1 2 3 4 5 6 7 8 9 10 11 12)
(1 3)(4 12)(5 11)(6 10)(7 9)
G:=sub<Sym(12)| (1,9,5)(2,6,10)(3,11,7)(4,8,12), (1,2,3,4,5,6,7,8,9,10,11,12), (1,3)(4,12)(5,11)(6,10)(7,9)>;
G:=Group( (1,9,5)(2,6,10)(3,11,7)(4,8,12), (1,2,3,4,5,6,7,8,9,10,11,12), (1,3)(4,12)(5,11)(6,10)(7,9) );
G=PermutationGroup([[(1,9,5),(2,6,10),(3,11,7),(4,8,12)], [(1,2,3,4,5,6,7,8,9,10,11,12)], [(1,3),(4,12),(5,11),(6,10),(7,9)]])
G:=TransitiveGroup(12,38);
►On 24 points - transitive group 24T74
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 17 21)(14 22 18)(15 19 23)(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)(2,10,6)(3,7,11)(4,12,8)(13,17,21)(14,22,18)(15,19,23)(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)(2,10,6)(3,7,11)(4,12,8)(13,17,21)(14,22,18)(15,19,23)(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),(2,10,6),(3,7,11),(4,12,8),(13,17,21),(14,22,18),(15,19,23),(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(24,74);
C3⋊D12 is a maximal subgroup of
D12⋊S3 D6.D6 D6.6D6 S3×D12 D6.3D6 S3×C3⋊D4 Dic3⋊D6 C3⋊D36 C9⋊D12 He3⋊2D4 He3⋊3D4 C33⋊7D4 C33⋊8D4 C33⋊9D4 GL2(𝔽3)⋊S3 D6.2S4 Dic3⋊S4 A4⋊D12 C3⋊D60 D6⋊2D15 D30⋊S3
C3⋊D12 is a maximal quotient of
C3⋊D24 D12.S3 C32⋊5SD16 C32⋊3Q16 D6⋊Dic3 C6.D12 Dic3⋊Dic3 C3⋊D36 C9⋊D12 He3⋊3D4 C33⋊7D4 C33⋊8D4 C33⋊9D4 Dic3⋊S4 A4⋊D12 C3⋊D60 D6⋊2D15 D30⋊S3
| action | f(x) | Disc(f) |
|---|---|---|
| 12T38 | x12-36x10+486x8-3084x6+9585x4-13608x2+6534 | 235·339·116·296 |
Matrix representation of C3⋊D12 ►in GL4(ℤ) generated by
| 0 | -1 | 0 | 0 |
| 1 | -1 | 0 | 0 |
| 0 | 0 | -1 | 1 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | -1 |
| 0 | 1 | 0 | 0 |
| -1 | 1 | 0 | 0 |
| -1 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | -1 |
| 0 | 0 | 0 | -1 |
G:=sub<GL(4,Integers())| [0,1,0,0,-1,-1,0,0,0,0,-1,-1,0,0,1,0],[0,0,0,-1,0,0,1,1,0,1,0,0,-1,-1,0,0],[-1,0,0,0,1,1,0,0,0,0,1,0,0,0,-1,-1] >;
C3⋊D12 in GAP, Magma, Sage, TeX
C_3\rtimes D_{12} % in TeX
G:=Group("C3:D12"); // GroupNames label
G:=SmallGroup(72,23);
// by ID
G=gap.SmallGroup(72,23);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,61,26,168,1204]);
// Polycyclic
G:=Group<a,b,c|a^3=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 C3⋊D12 in TeX
Character table of C3⋊D12 in TeX