direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3xD4, C4:C6, C12:3C2, C22:2C6, C6.6C22, (C2xC6):1C2, C2.1(C2xC6), SmallGroup(24,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xD4
G = < a,b,c | a3=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C3, C22, C6, D4, C2xC6, C3xD4
Character table of C3xD4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ3 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ32 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ32 | linear of order 6 |
| ρ8 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | ζ32 | ζ3 | ζ6 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ10 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | ζ3 | ζ32 | ζ65 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ3 | ζ6 | ζ65 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ15 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 12 points - transitive group 12T14
Generators in S12
(1 7 9)(2 8 10)(3 5 11)(4 6 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)
G:=sub<Sym(12)| (1,7,9)(2,8,10)(3,5,11)(4,6,12), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)>;
G:=Group( (1,7,9)(2,8,10)(3,5,11)(4,6,12), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11) );
G=PermutationGroup([[(1,7,9),(2,8,10),(3,5,11),(4,6,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]])
G:=TransitiveGroup(12,14);
►Regular action on 24 points - transitive group 24T15
Generators in S24
(1 19 15)(2 20 16)(3 17 13)(4 18 14)(5 10 21)(6 11 22)(7 12 23)(8 9 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 9)(2 12)(3 11)(4 10)(5 14)(6 13)(7 16)(8 15)(17 22)(18 21)(19 24)(20 23)
G:=sub<Sym(24)| (1,19,15)(2,20,16)(3,17,13)(4,18,14)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9)(2,12)(3,11)(4,10)(5,14)(6,13)(7,16)(8,15)(17,22)(18,21)(19,24)(20,23)>;
G:=Group( (1,19,15)(2,20,16)(3,17,13)(4,18,14)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9)(2,12)(3,11)(4,10)(5,14)(6,13)(7,16)(8,15)(17,22)(18,21)(19,24)(20,23) );
G=PermutationGroup([[(1,19,15),(2,20,16),(3,17,13),(4,18,14),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,9),(2,12),(3,11),(4,10),(5,14),(6,13),(7,16),(8,15),(17,22),(18,21),(19,24),(20,23)]])
G:=TransitiveGroup(24,15);
C3xD4 is a maximal subgroup of
D4:S3 D4.S3 D4:2S3 D4.A4 C4:F7 Dic7:C6 D52:C3 D26:C6 D76:C3 D38:C6
C3xD4 is a maximal quotient of
C4:F7 Dic7:C6 D52:C3 D26:C6 D76:C3 D38:C6
| action | f(x) | Disc(f) |
|---|---|---|
| 12T14 | x12-24x10+226x8-1056x6+2552x4-3008x2+1352 | 257·710·132 |
Matrix representation of C3xD4 ►in GL2(F7) generated by
| 4 | 0 |
| 0 | 4 |
| 0 | 6 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(7))| [4,0,0,4],[0,1,6,0],[0,1,1,0] >;
C3xD4 in GAP, Magma, Sage, TeX
C_3\times D_4
% in TeX
G:=Group("C3xD4"); // GroupNames label
G:=SmallGroup(24,10);
// by ID
G=gap.SmallGroup(24,10);
# by ID
G:=PCGroup([4,-2,-2,-3,-2,113]);
// Polycyclic
G:=Group<a,b,c|a^3=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C3xD4 in TeX
Character table of C3xD4 in TeX