metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.11D4, C23.2S3, C22.7D6, C22⋊2Dic3, (C2×C6)⋊2C4, C6.9(C2×C4), C3⋊2(C22⋊C4), (C2×Dic3)⋊2C2, C2.3(C3⋊D4), (C22×C6).2C2, (C2×C6).7C22, C2.5(C2×Dic3), SmallGroup(48,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.D4
G = < a,b,c | a6=b4=1, c2=a3, bab-1=cac-1=a-1, cbc-1=a3b-1 >
Character table of C6.D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 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 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | -i | -i | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 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 | 0 | 0 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -√-3 | √-3 | -1 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | √-3 | -√-3 | 1 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | √-3 | -√-3 | -1 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -√-3 | √-3 | 1 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
►On 24 points - transitive group 24T44
(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 23 14 10)(2 22 15 9)(3 21 16 8)(4 20 17 7)(5 19 18 12)(6 24 13 11)
(1 20 4 23)(2 19 5 22)(3 24 6 21)(7 17 10 14)(8 16 11 13)(9 15 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), (1,23,14,10)(2,22,15,9)(3,21,16,8)(4,20,17,7)(5,19,18,12)(6,24,13,11), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,17,10,14)(8,16,11,13)(9,15,12,18)>;
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), (1,23,14,10)(2,22,15,9)(3,21,16,8)(4,20,17,7)(5,19,18,12)(6,24,13,11), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,17,10,14)(8,16,11,13)(9,15,12,18) );
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)], [(1,23,14,10),(2,22,15,9),(3,21,16,8),(4,20,17,7),(5,19,18,12),(6,24,13,11)], [(1,20,4,23),(2,19,5,22),(3,24,6,21),(7,17,10,14),(8,16,11,13),(9,15,12,18)]])
G:=TransitiveGroup(24,44);
C6.D4 is a maximal subgroup of
C23.6D6 C23.7D6 C23.16D6 Dic3.D4 C23.8D6 S3×C22⋊C4 C23.9D6 C23.11D6 C12.48D4 C23.26D6 C4×C3⋊D4 C23.28D6 D4×Dic3 C23.23D6 C23.12D6 C23⋊2D6 D6⋊3D4 C23.14D6 C24⋊4S3 C18.D4 C6.S4 D6⋊Dic3 C62⋊5C4 C6.7S4 C23.14S4 C23.15S4 C25.S3 D10⋊Dic3 C30.38D4 D10.D6 D14⋊Dic3 C42.38D4 S32⋊Dic3 C62⋊11Dic3
C6.D4 is a maximal quotient of
C12.55D4 C6.C42 D4⋊Dic3 C12.D4 C23.7D6 Q8⋊2Dic3 C12.10D4 Q8⋊3Dic3 C18.D4 D6⋊Dic3 C62⋊5C4 C25.S3 D10⋊Dic3 C30.38D4 D10.D6 D14⋊Dic3 C42.38D4 S32⋊Dic3 C62⋊11Dic3
Matrix representation of C6.D4 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 10 |
| 5 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 8 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(13))| [12,0,0,0,4,0,0,0,10],[5,0,0,0,0,1,0,1,0],[8,0,0,0,0,12,0,1,0] >;
C6.D4 in GAP, Magma, Sage, TeX
C_6.D_4
% in TeX
G:=Group("C6.D4"); // GroupNames label
G:=SmallGroup(48,19);
// by ID
G=gap.SmallGroup(48,19);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,20,101,804]);
// Polycyclic
G:=Group<a,b,c|a^6=b^4=1,c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations
Export
Subgroup lattice of C6.D4 in TeX
Character table of C6.D4 in TeX