metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:C4, C6.6D4, C2.2D12, C22.6D6, (C2xC4):1S3, (C2xC12):1C2, C2.5(C4xS3), C6.4(C2xC4), C3:1(C22:C4), (C22xS3).C2, (C2xDic3):1C2, C2.2(C3:D4), (C2xC6).6C22, SmallGroup(48,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6:C4
G = < a,b,c | a6=b2=c4=1, bab=a-1, ac=ca, cbc-1=a3b >
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, C4xS3, D12, C3:D4, D6:C4
Character table of D6:C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 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 | 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 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | -1 | 1 | 1 | -i | -i | i | i | complex lifted from C4xS3 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | -1 | 1 | 1 | i | i | -i | -i | complex lifted from C4xS3 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -√-3 | √-3 | complex lifted from C3:D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | √-3 | -√-3 | complex lifted from C3:D4 |
►On 24 points - transitive group 24T33
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 21)(14 20)(15 19)(16 24)(17 23)(18 22)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)
G:=sub<Sym(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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18)]])
G:=TransitiveGroup(24,33);
D6:C4 is a maximal subgroup of
C42:2S3 C4xD12 C42:7S3 C42:3S3 S3xC22:C4 Dic3:4D4 D6:D4 C23.9D6 Dic3:D4 C23.11D6 C23.21D6 C4:C4:7S3 Dic3:5D4 D6.D4 C12:D4 D6:Q8 C4.D12 C4:C4:S3 C4xC3:D4 C23.28D6 C12:7D4 C23:2D6 C23.14D6 D6:3Q8 C12.23D4 D18:C4 D6:Dic3 C6.D12 C6.11D12 GL2(F3):C4 Q8.2D12 C24.5D6 SL2(F3):D4 D6:Dic5 D30:4C4 D30:3C4 D6:F5 D6:Dic7 D42:C4 C2.D84 C32:D6:C4 D6:(C32:C4) C3:S3.2D12 (C3xC6).8D12
D6:C4 is a maximal quotient of
C42:4S3 C23.6D6 C6.D8 C6.SD16 C2.Dic12 D6:C8 C2.D24 C12.46D4 C12.47D4 D12:C4 C6.C42 D18:C4 D6:Dic3 C6.D12 C6.11D12 C24.5D6 D6:Dic5 D30:4C4 D30:3C4 D6:F5 D6:Dic7 D42:C4 C2.D84 D6:(C32:C4) C3:S3.2D12 (C3xC6).8D12
Matrix representation of D6:C4 ►in GL3(F13) generated by
| 1 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 12 | 0 |
| 12 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 12 |
| 5 | 0 | 0 |
| 0 | 11 | 9 |
| 0 | 4 | 2 |
G:=sub<GL(3,GF(13))| [1,0,0,0,1,12,0,1,0],[12,0,0,0,1,0,0,1,12],[5,0,0,0,11,4,0,9,2] >;
D6:C4 in GAP, Magma, Sage, TeX
D_6\rtimes C_4
% in TeX
G:=Group("D6:C4"); // GroupNames label
G:=SmallGroup(48,14);
// by ID
G=gap.SmallGroup(48,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,101,26,804]);
// Polycyclic
G:=Group<a,b,c|a^6=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations
Export
Subgroup lattice of D6:C4 in TeX
Character table of D6:C4 in TeX