direct product, metabelian, soluble, monomial, A-group
Aliases: C3×A4, C22⋊C32, (C2×C6)⋊C3, SmallGroup(36,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C3×A4 |
Generators and relations for C3×A4
G = < a,b,c,d | a3=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Character table of C3×A4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | |
| size | 1 | 3 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | linear of order 3 |
| ρ8 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 3 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from A4 |
| ρ11 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | complex faithful |
| ρ12 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | complex faithful |
►On 12 points - transitive group 12T20
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 4)(2 5)(3 6)(7 12)(8 10)(9 11)
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(1 2 3)(4 11 7)(5 12 8)(6 10 9)
G:=sub<Sym(12)| (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,4)(2,5)(3,6)(7,12)(8,10)(9,11), (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,2,3)(4,11,7)(5,12,8)(6,10,9)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,4)(2,5)(3,6)(7,12)(8,10)(9,11), (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,2,3)(4,11,7)(5,12,8)(6,10,9) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,4),(2,5),(3,6),(7,12),(8,10),(9,11)], [(1,10),(2,11),(3,12),(4,8),(5,9),(6,7)], [(1,2,3),(4,11,7),(5,12,8),(6,10,9)]])
G:=TransitiveGroup(12,20);
►On 18 points - transitive group 18T8
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(4 16)(5 17)(6 18)(10 13)(11 14)(12 15)
(1 7)(2 8)(3 9)(10 13)(11 14)(12 15)
(1 16 10)(2 17 11)(3 18 12)(4 13 7)(5 14 8)(6 15 9)
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15), (1,16,10)(2,17,11)(3,18,12)(4,13,7)(5,14,8)(6,15,9)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15), (1,16,10)(2,17,11)(3,18,12)(4,13,7)(5,14,8)(6,15,9) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(4,16),(5,17),(6,18),(10,13),(11,14),(12,15)], [(1,7),(2,8),(3,9),(10,13),(11,14),(12,15)], [(1,16,10),(2,17,11),(3,18,12),(4,13,7),(5,14,8),(6,15,9)]])
G:=TransitiveGroup(18,8);
C3×A4 is a maximal subgroup of
C3⋊S4 C9⋊A4 C32⋊A4
C3×A4 is a maximal quotient of C9⋊A4 C32.A4 C32⋊A4
| action | f(x) | Disc(f) |
|---|---|---|
| 12T20 | x12-3x11-30x10+73x9+303x8-603x7-1171x6+2034x5+1254x4-2511x3+420x2+228x+8 | 224·316·76·1636·61257412 |
Matrix representation of C3×A4 ►in GL3(𝔽7) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 0 | 6 | 1 |
| 0 | 6 | 0 |
| 1 | 6 | 0 |
| 6 | 0 | 0 |
| 6 | 0 | 1 |
| 6 | 1 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(7))| [4,0,0,0,4,0,0,0,4],[0,0,1,6,6,6,1,0,0],[6,6,6,0,0,1,0,1,0],[0,0,1,1,0,0,0,1,0] >;
C3×A4 in GAP, Magma, Sage, TeX
C_3\times A_4
% in TeX
G:=Group("C3xA4"); // GroupNames label
G:=SmallGroup(36,11);
// by ID
G=gap.SmallGroup(36,11);
# by ID
G:=PCGroup([4,-3,-3,-2,2,218,435]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C3×A4 in TeX
Character table of C3×A4 in TeX