direct product, metabelian, soluble, monomial, A-group
Aliases: S3×A4, C3⋊(C2×A4), (C2×C6)⋊C6, (C3×A4)⋊3C2, (C22×S3)⋊C3, C22⋊2(C3×S3), SmallGroup(72,44)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C6 — S3×A4 |
Generators and relations for S3×A4
G = < a,b,c,d,e | a3=b2=c2=d2=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Character table of S3×A4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | |
| size | 1 | 3 | 3 | 9 | 2 | 4 | 4 | 8 | 8 | 6 | 12 | 12 | |
| 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 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | linear of order 6 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | 0 | complex lifted from C3×S3 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | 0 | complex lifted from C3×S3 |
| ρ10 | 3 | -1 | -3 | 1 | 3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ11 | 3 | -1 | 3 | -1 | 3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ12 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | orthogonal faithful |
►On 12 points - transitive group 12T43
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(2 3)(5 6)(8 9)(11 12)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)
(4 7 10)(5 8 11)(6 9 12)
G:=sub<Sym(12)| (1,2,3)(4,5,6)(7,8,9)(10,11,12), (2,3)(5,6)(8,9)(11,12), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (4,7,10)(5,8,11)(6,9,12)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12), (2,3)(5,6)(8,9)(11,12), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (4,7,10)(5,8,11)(6,9,12) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(2,3),(5,6),(8,9),(11,12)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)], [(4,7,10),(5,8,11),(6,9,12)]])
G:=TransitiveGroup(12,43);
►On 18 points - transitive group 18T31
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(2 3)(5 6)(8 9)(11 12)(14 15)(17 18)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (2,3)(5,6)(8,9)(11,12)(14,15)(17,18), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (2,3)(5,6)(8,9)(11,12)(14,15)(17,18), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(2,3),(5,6),(8,9),(11,12),(14,15),(17,18)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12)]])
G:=TransitiveGroup(18,31);
►On 18 points - transitive group 18T32
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 4)(2 6)(3 5)(7 10)(8 12)(9 11)(13 16)(14 18)(15 17)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,4)(2,6)(3,5)(7,10)(8,12)(9,11)(13,16)(14,18)(15,17), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,4)(2,6)(3,5)(7,10)(8,12)(9,11)(13,16)(14,18)(15,17), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,4),(2,6),(3,5),(7,10),(8,12),(9,11),(13,16),(14,18),(15,17)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12)]])
G:=TransitiveGroup(18,32);
►On 24 points - transitive group 24T78
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 14)(2 13)(3 15)(4 17)(5 16)(6 18)(7 20)(8 19)(9 21)(10 23)(11 22)(12 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(4 7 10)(5 8 11)(6 9 12)(16 19 22)(17 20 23)(18 21 24)
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), (1,14)(2,13)(3,15)(4,17)(5,16)(6,18)(7,20)(8,19)(9,21)(10,23)(11,22)(12,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (4,7,10)(5,8,11)(6,9,12)(16,19,22)(17,20,23)(18,21,24)>;
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), (1,14)(2,13)(3,15)(4,17)(5,16)(6,18)(7,20)(8,19)(9,21)(10,23)(11,22)(12,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (4,7,10)(5,8,11)(6,9,12)(16,19,22)(17,20,23)(18,21,24) );
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)], [(1,14),(2,13),(3,15),(4,17),(5,16),(6,18),(7,20),(8,19),(9,21),(10,23),(11,22),(12,24)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(4,7,10),(5,8,11),(6,9,12),(16,19,22),(17,20,23),(18,21,24)]])
G:=TransitiveGroup(24,78);
►On 24 points - transitive group 24T83
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 14)(2 13)(3 15)(4 17)(5 16)(6 18)(7 20)(8 19)(9 21)(10 23)(11 22)(12 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 2 3)(4 8 12)(5 9 10)(6 7 11)(13 15 14)(16 21 23)(17 19 24)(18 20 22)
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), (1,14)(2,13)(3,15)(4,17)(5,16)(6,18)(7,20)(8,19)(9,21)(10,23)(11,22)(12,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,8,12)(5,9,10)(6,7,11)(13,15,14)(16,21,23)(17,19,24)(18,20,22)>;
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), (1,14)(2,13)(3,15)(4,17)(5,16)(6,18)(7,20)(8,19)(9,21)(10,23)(11,22)(12,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,8,12)(5,9,10)(6,7,11)(13,15,14)(16,21,23)(17,19,24)(18,20,22) );
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)], [(1,14),(2,13),(3,15),(4,17),(5,16),(6,18),(7,20),(8,19),(9,21),(10,23),(11,22),(12,24)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3),(4,8,12),(5,9,10),(6,7,11),(13,15,14),(16,21,23),(17,19,24),(18,20,22)]])
G:=TransitiveGroup(24,83);
S3×A4 is a maximal subgroup of
D9⋊A4 C62⋊C6 (C4×C12)⋊C6 C42⋊C3⋊S3 (C22×S3)⋊A4
S3×A4 is a maximal quotient of Dic3.A4 D9⋊A4 C62⋊C6 (C4×C12)⋊C6 C42⋊C3⋊S3 (C22×S3)⋊A4
| action | f(x) | Disc(f) |
|---|---|---|
| 12T43 | x12-6x11+18x10-28x9+18x8+36x5+32x3+24x2+4 | 234·324·965892 |
Matrix representation of S3×A4 ►in GL5(𝔽7)
| 0 | 6 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 6 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 6 | 6 | 6 |
| 0 | 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 6 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(7))| [0,1,0,0,0,6,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,1,0,6,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,6,0,0,0,1,6,0],[4,0,0,0,0,0,4,0,0,0,0,0,1,6,0,0,0,0,6,1,0,0,0,6,0] >;
S3×A4 in GAP, Magma, Sage, TeX
S_3\times A_4
% in TeX
G:=Group("S3xA4"); // GroupNames label
G:=SmallGroup(72,44);
// by ID
G=gap.SmallGroup(72,44);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-3,142,68,1204]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^2=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export
Subgroup lattice of S3×A4 in TeX
Character table of S3×A4 in TeX