direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic3, C6⋊C4, C2.2D6, C22.S3, C6.4C22, C3⋊2(C2×C4), (C2×C6).C2, SmallGroup(24,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C2×Dic3 |
Generators and relations for C2×Dic3
G = < a,b,c | a2=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C2×Dic3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | i | i | -i | -i | -1 | -1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | -1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -i | -i | i | i | -1 | -1 | 1 | linear of order 4 |
| ρ9 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ12 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
►Regular action on 24 points - transitive group 24T6
Generators in S24
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 16 4 13)(2 15 5 18)(3 14 6 17)(7 20 10 23)(8 19 11 22)(9 24 12 21)
G:=sub<Sym(24)| (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,4,13)(2,15,5,18)(3,14,6,17)(7,20,10,23)(8,19,11,22)(9,24,12,21)>;
G:=Group( (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,4,13)(2,15,5,18)(3,14,6,17)(7,20,10,23)(8,19,11,22)(9,24,12,21) );
G=PermutationGroup([[(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,16,4,13),(2,15,5,18),(3,14,6,17),(7,20,10,23),(8,19,11,22),(9,24,12,21)]])
G:=TransitiveGroup(24,6);
C2×Dic3 is a maximal subgroup of
Dic3⋊C4 C4⋊Dic3 D6⋊C4 C6.D4 S3×C2×C4 D4⋊2S3 Q8⋊Dic3
C2×Dic3 is a maximal quotient of C4.Dic3 C4⋊Dic3 C6.D4
Matrix representation of C2×Dic3 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 1 | 12 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 5 | 8 |
| 0 | 0 | 8 |
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[1,0,0,0,1,1,0,12,0],[1,0,0,0,5,0,0,8,8] >;
C2×Dic3 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_3 % in TeX
G:=Group("C2xDic3"); // GroupNames label
G:=SmallGroup(24,7);
// by ID
G=gap.SmallGroup(24,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-3,16,259]);
// Polycyclic
G:=Group<a,b,c|a^2=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×Dic3 in TeX
Character table of C2×Dic3 in TeX