direct product, metabelian, soluble, monomial, A-group
Aliases: C6×A4, C23⋊C32, (C2×C6)⋊2C6, C22⋊(C3×C6), (C22×C6)⋊C3, SmallGroup(72,47)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C6×A4 |
Generators and relations for C6×A4
G = < a,b,c,d | a6=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Character table of C6×A4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | -1 | -1 | -1 | 1 | 1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ5 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ65 | ζ6 | ζ65 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ9 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ65 | ζ6 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | -1 | linear of order 6 |
| ρ10 | 1 | -1 | -1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | -1 | -1 | -1 | 1 | 1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ12 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ6 | ζ65 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | -1 | linear of order 6 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ14 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ32 | ζ3 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | linear of order 6 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ16 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ6 | ζ65 | ζ6 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ6 | linear of order 6 |
| ρ17 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | linear of order 3 |
| ρ18 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ3 | ζ32 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | linear of order 6 |
| ρ19 | 3 | -3 | 1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ20 | 3 | 3 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ21 | 3 | -3 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | ζ3 | ζ6 | ζ65 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 3 | -3 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3/2 | 3-3√-3/2 | ζ32 | ζ65 | ζ6 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 3 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | ζ6 | ζ65 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A4 |
| ρ24 | 3 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | ζ65 | ζ6 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A4 |
►On 18 points - transitive group 18T25
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 15 11)(2 16 12)(3 17 7)(4 18 8)(5 13 9)(6 14 10)
G:=sub<Sym(18)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,15,11)(2,16,12)(3,17,7)(4,18,8)(5,13,9)(6,14,10)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,15,11)(2,16,12)(3,17,7)(4,18,8)(5,13,9)(6,14,10) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)], [(1,15,11),(2,16,12),(3,17,7),(4,18,8),(5,13,9),(6,14,10)]])
G:=TransitiveGroup(18,25);
►On 24 points - transitive group 24T71
Generators in S24
(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 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)
(1 5 3)(2 6 4)(7 23 13)(8 24 14)(9 19 15)(10 20 16)(11 21 17)(12 22 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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16), (1,5,3)(2,6,4)(7,23,13)(8,24,14)(9,19,15)(10,20,16)(11,21,17)(12,22,18)>;
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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16), (1,5,3)(2,6,4)(7,23,13)(8,24,14)(9,19,15)(10,20,16)(11,21,17)(12,22,18) );
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)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,7),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,19),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16)], [(1,5,3),(2,6,4),(7,23,13),(8,24,14),(9,19,15),(10,20,16),(11,21,17),(12,22,18)]])
G:=TransitiveGroup(24,71);
C6×A4 is a maximal subgroup of
C6.7S4
Matrix representation of C6×A4 ►in GL3(𝔽7) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 6 | 0 | 0 |
| 4 | 0 | 1 |
| 4 | 1 | 0 |
| 0 | 5 | 2 |
| 3 | 0 | 6 |
| 0 | 0 | 6 |
| 2 | 0 | 4 |
| 0 | 0 | 5 |
| 0 | 2 | 5 |
G:=sub<GL(3,GF(7))| [3,0,0,0,3,0,0,0,3],[6,4,4,0,0,1,0,1,0],[0,3,0,5,0,0,2,6,6],[2,0,0,0,0,2,4,5,5] >;
C6×A4 in GAP, Magma, Sage, TeX
C_6\times A_4
% in TeX
G:=Group("C6xA4"); // GroupNames label
G:=SmallGroup(72,47);
// by ID
G=gap.SmallGroup(72,47);
# by ID
G:=PCGroup([5,-2,-3,-3,-2,2,368,684]);
// Polycyclic
G:=Group<a,b,c,d|a^6=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 C6×A4 in TeX
Character table of C6×A4 in TeX