metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6, C3⋊Q8, C4.S3, C2.3D6, C12.1C2, Dic3.C2, C6.1C22, SmallGroup(24,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6
G = < a,b | a12=1, b2=a6, bab-1=a-1 >
Character table of Dic6
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 12A | 12B | |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ8 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | √3 | -√3 | symplectic faithful, Schur index 2 |
| ρ9 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | -√3 | √3 | symplectic faithful, Schur index 2 |
►Regular action on 24 points - transitive group 24T5
(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 17 7 23)(2 16 8 22)(3 15 9 21)(4 14 10 20)(5 13 11 19)(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), (1,17,7,23)(2,16,8,22)(3,15,9,21)(4,14,10,20)(5,13,11,19)(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), (1,17,7,23)(2,16,8,22)(3,15,9,21)(4,14,10,20)(5,13,11,19)(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)], [(1,17,7,23),(2,16,8,22),(3,15,9,21),(4,14,10,20),(5,13,11,19),(6,24,12,18)]])
G:=TransitiveGroup(24,5);
Dic6 is a maximal subgroup of
A4⋊Q8 C4.S4 C33⋊Q8 CSU2(𝔽5)
Dic6p: Dic12 Dic18 Dic30 Dic42 Dic66 Dic78 Dic102 Dic114 ...
C2p.D6: C24⋊C2 D4.S3 C3⋊Q16 C4○D12 D4⋊2S3 S3×Q8 C32⋊2Q8 C32⋊4Q8 ...
Dic6 is a maximal quotient of
A4⋊Q8 C33⋊Q8
C6.D2p: Dic3⋊C4 C4⋊Dic3 Dic18 C32⋊2Q8 C32⋊4Q8 C15⋊Q8 Dic30 C21⋊Q8 ...
Matrix representation of Dic6 ►in GL2(𝔽11) generated by
| 2 | 7 |
| 7 | 3 |
| 0 | 10 |
| 1 | 0 |
G:=sub<GL(2,GF(11))| [2,7,7,3],[0,1,10,0] >;
Dic6 in GAP, Magma, Sage, TeX
{\rm Dic}_6 % in TeX
G:=Group("Dic6"); // GroupNames label
G:=SmallGroup(24,4);
// by ID
G=gap.SmallGroup(24,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-3,16,49,21,259]);
// Polycyclic
G:=Group<a,b|a^12=1,b^2=a^6,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic6 in TeX
Character table of Dic6 in TeX