direct product, non-abelian, soluble, monomial
Aliases: A4×S4, A42⋊2C2, A4⋊(C2×A4), C22⋊(S3×A4), (C22×S4)⋊C3, (C22×A4)⋊C6, C22⋊2(C3×S4), C24⋊1(C3×S3), (C22×A4)⋊1S3, SmallGroup(288,1024)
Series: Derived ►Chief ►Lower central ►Upper central
| C22×A4 — A4×S4 |
Generators and relations for A4×S4
G = < a,b,c,d,e,f,g | a2=b2=c3=d2=e2=f3=g2=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, ag=ga, cbc-1=a, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >
Subgroups: 644 in 86 conjugacy classes, 12 normal (all characteristic)
C1, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, C12, A4, A4, D6, C2×C6, C22×C4, C2×D4, C24, C24, C3×S3, C3×D4, S4, S4, C2×A4, C22×S3, C22×D4, C3×A4, C4×A4, C2×S4, C22×A4, C22×A4, C22⋊A4, C3×S4, S3×A4, D4×A4, C22×S4, A42, A4×S4
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, S4, C2×A4, C3×S4, S3×A4, A4×S4
Character table of A4×S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 12A | 12B | |
| size | 1 | 3 | 3 | 6 | 9 | 18 | 4 | 4 | 8 | 32 | 32 | 6 | 18 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | -1 | -1 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | -1 | -1 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 0 | 2 | 0 | -1+√-3 | -1-√-3 | -1 | ζ6 | ζ65 | 0 | 0 | -1+√-3 | -1-√-3 | -1 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ9 | 2 | 2 | 2 | 0 | 2 | 0 | -1-√-3 | -1+√-3 | -1 | ζ65 | ζ6 | 0 | 0 | -1-√-3 | -1+√-3 | -1 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ10 | 3 | -1 | 3 | -3 | -1 | 1 | 0 | 0 | 3 | 0 | 0 | -3 | 1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ11 | 3 | -1 | 3 | 3 | -1 | -1 | 0 | 0 | 3 | 0 | 0 | 3 | -1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ12 | 3 | 3 | -1 | 1 | -1 | 1 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | -1 | -1 | orthogonal lifted from S4 |
| ρ13 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | -1 | 1 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | 0 | ζ3 | ζ32 | ζ65 | ζ6 | complex lifted from C3×S4 |
| ρ15 | 3 | 3 | -1 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1 | 1 | ζ6 | ζ65 | 0 | ζ65 | ζ6 | ζ3 | ζ32 | complex lifted from C3×S4 |
| ρ16 | 3 | 3 | -1 | 1 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | 0 | ζ32 | ζ3 | ζ6 | ζ65 | complex lifted from C3×S4 |
| ρ17 | 3 | 3 | -1 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1 | 1 | ζ65 | ζ6 | 0 | ζ6 | ζ65 | ζ32 | ζ3 | complex lifted from C3×S4 |
| ρ18 | 6 | -2 | 6 | 0 | -2 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×A4 |
| ρ19 | 9 | -3 | -3 | 3 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ20 | 9 | -3 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 16 points - transitive group 16T709
Generators in S16
(1 8)(2 15)(3 11)(4 7)(5 6)(9 10)(12 13)(14 16)
(1 9)(2 16)(3 12)(4 5)(6 7)(8 10)(11 13)(14 15)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 3)(2 4)(5 16)(6 14)(7 15)(8 11)(9 12)(10 13)
(1 4)(2 3)(5 9)(6 10)(7 8)(11 15)(12 16)(13 14)
(2 3 4)(5 16 12)(6 14 13)(7 15 11)
(2 3)(11 15)(12 16)(13 14)
G:=sub<Sym(16)| (1,8)(2,15)(3,11)(4,7)(5,6)(9,10)(12,13)(14,16), (1,9)(2,16)(3,12)(4,5)(6,7)(8,10)(11,13)(14,15), (5,6,7)(8,9,10)(11,12,13)(14,15,16), (1,3)(2,4)(5,16)(6,14)(7,15)(8,11)(9,12)(10,13), (1,4)(2,3)(5,9)(6,10)(7,8)(11,15)(12,16)(13,14), (2,3,4)(5,16,12)(6,14,13)(7,15,11), (2,3)(11,15)(12,16)(13,14)>;
G:=Group( (1,8)(2,15)(3,11)(4,7)(5,6)(9,10)(12,13)(14,16), (1,9)(2,16)(3,12)(4,5)(6,7)(8,10)(11,13)(14,15), (5,6,7)(8,9,10)(11,12,13)(14,15,16), (1,3)(2,4)(5,16)(6,14)(7,15)(8,11)(9,12)(10,13), (1,4)(2,3)(5,9)(6,10)(7,8)(11,15)(12,16)(13,14), (2,3,4)(5,16,12)(6,14,13)(7,15,11), (2,3)(11,15)(12,16)(13,14) );
G=PermutationGroup([[(1,8),(2,15),(3,11),(4,7),(5,6),(9,10),(12,13),(14,16)], [(1,9),(2,16),(3,12),(4,5),(6,7),(8,10),(11,13),(14,15)], [(5,6,7),(8,9,10),(11,12,13),(14,15,16)], [(1,3),(2,4),(5,16),(6,14),(7,15),(8,11),(9,12),(10,13)], [(1,4),(2,3),(5,9),(6,10),(7,8),(11,15),(12,16),(13,14)], [(2,3,4),(5,16,12),(6,14,13),(7,15,11)], [(2,3),(11,15),(12,16),(13,14)]])
G:=TransitiveGroup(16,709);
►On 18 points - transitive group 18T114
Generators in S18
(2 18)(3 16)(4 7)(5 8)(11 13)(12 14)
(1 17)(3 16)(5 8)(6 9)(10 15)(12 14)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 17)(2 18)(3 16)(4 7)(5 8)(6 9)
(4 7)(5 8)(6 9)(10 15)(11 13)(12 14)
(1 9 10)(2 7 11)(3 8 12)(4 13 18)(5 14 16)(6 15 17)
(1 17)(2 18)(3 16)(4 11)(5 12)(6 10)(7 13)(8 14)(9 15)
G:=sub<Sym(18)| (2,18)(3,16)(4,7)(5,8)(11,13)(12,14), (1,17)(3,16)(5,8)(6,9)(10,15)(12,14), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,17)(2,18)(3,16)(4,7)(5,8)(6,9), (4,7)(5,8)(6,9)(10,15)(11,13)(12,14), (1,9,10)(2,7,11)(3,8,12)(4,13,18)(5,14,16)(6,15,17), (1,17)(2,18)(3,16)(4,11)(5,12)(6,10)(7,13)(8,14)(9,15)>;
G:=Group( (2,18)(3,16)(4,7)(5,8)(11,13)(12,14), (1,17)(3,16)(5,8)(6,9)(10,15)(12,14), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,17)(2,18)(3,16)(4,7)(5,8)(6,9), (4,7)(5,8)(6,9)(10,15)(11,13)(12,14), (1,9,10)(2,7,11)(3,8,12)(4,13,18)(5,14,16)(6,15,17), (1,17)(2,18)(3,16)(4,11)(5,12)(6,10)(7,13)(8,14)(9,15) );
G=PermutationGroup([[(2,18),(3,16),(4,7),(5,8),(11,13),(12,14)], [(1,17),(3,16),(5,8),(6,9),(10,15),(12,14)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,17),(2,18),(3,16),(4,7),(5,8),(6,9)], [(4,7),(5,8),(6,9),(10,15),(11,13),(12,14)], [(1,9,10),(2,7,11),(3,8,12),(4,13,18),(5,14,16),(6,15,17)], [(1,17),(2,18),(3,16),(4,11),(5,12),(6,10),(7,13),(8,14),(9,15)]])
G:=TransitiveGroup(18,114);
►On 18 points - transitive group 18T115
Generators in S18
(2 18)(3 16)(4 7)(5 8)(11 13)(12 14)
(1 17)(3 16)(5 8)(6 9)(10 15)(12 14)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 17)(2 18)(3 16)(4 7)(5 8)(6 9)
(4 7)(5 8)(6 9)(10 15)(11 13)(12 14)
(1 9 10)(2 7 11)(3 8 12)(4 13 18)(5 14 16)(6 15 17)
(4 13)(5 14)(6 15)(7 11)(8 12)(9 10)
G:=sub<Sym(18)| (2,18)(3,16)(4,7)(5,8)(11,13)(12,14), (1,17)(3,16)(5,8)(6,9)(10,15)(12,14), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,17)(2,18)(3,16)(4,7)(5,8)(6,9), (4,7)(5,8)(6,9)(10,15)(11,13)(12,14), (1,9,10)(2,7,11)(3,8,12)(4,13,18)(5,14,16)(6,15,17), (4,13)(5,14)(6,15)(7,11)(8,12)(9,10)>;
G:=Group( (2,18)(3,16)(4,7)(5,8)(11,13)(12,14), (1,17)(3,16)(5,8)(6,9)(10,15)(12,14), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,17)(2,18)(3,16)(4,7)(5,8)(6,9), (4,7)(5,8)(6,9)(10,15)(11,13)(12,14), (1,9,10)(2,7,11)(3,8,12)(4,13,18)(5,14,16)(6,15,17), (4,13)(5,14)(6,15)(7,11)(8,12)(9,10) );
G=PermutationGroup([[(2,18),(3,16),(4,7),(5,8),(11,13),(12,14)], [(1,17),(3,16),(5,8),(6,9),(10,15),(12,14)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,17),(2,18),(3,16),(4,7),(5,8),(6,9)], [(4,7),(5,8),(6,9),(10,15),(11,13),(12,14)], [(1,9,10),(2,7,11),(3,8,12),(4,13,18),(5,14,16),(6,15,17)], [(4,13),(5,14),(6,15),(7,11),(8,12),(9,10)]])
G:=TransitiveGroup(18,115);
►On 24 points - transitive group 24T634
Generators in S24
(1 22)(2 21)(3 15)(4 11)(5 18)(6 9)(7 8)(10 12)(13 14)(16 17)(19 20)(23 24)
(1 23)(2 19)(3 13)(4 12)(5 16)(6 7)(8 9)(10 11)(14 15)(17 18)(20 21)(22 24)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 2)(3 4)(10 14)(11 15)(12 13)(19 23)(20 24)(21 22)
(3 4)(5 6)(7 16)(8 17)(9 18)(10 14)(11 15)(12 13)
(1 3 5)(2 4 6)(7 19 12)(8 20 10)(9 21 11)(13 16 23)(14 17 24)(15 18 22)
(3 5)(4 6)(7 12)(8 10)(9 11)(13 16)(14 17)(15 18)
G:=sub<Sym(24)| (1,22)(2,21)(3,15)(4,11)(5,18)(6,9)(7,8)(10,12)(13,14)(16,17)(19,20)(23,24), (1,23)(2,19)(3,13)(4,12)(5,16)(6,7)(8,9)(10,11)(14,15)(17,18)(20,21)(22,24), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,4)(10,14)(11,15)(12,13)(19,23)(20,24)(21,22), (3,4)(5,6)(7,16)(8,17)(9,18)(10,14)(11,15)(12,13), (1,3,5)(2,4,6)(7,19,12)(8,20,10)(9,21,11)(13,16,23)(14,17,24)(15,18,22), (3,5)(4,6)(7,12)(8,10)(9,11)(13,16)(14,17)(15,18)>;
G:=Group( (1,22)(2,21)(3,15)(4,11)(5,18)(6,9)(7,8)(10,12)(13,14)(16,17)(19,20)(23,24), (1,23)(2,19)(3,13)(4,12)(5,16)(6,7)(8,9)(10,11)(14,15)(17,18)(20,21)(22,24), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,4)(10,14)(11,15)(12,13)(19,23)(20,24)(21,22), (3,4)(5,6)(7,16)(8,17)(9,18)(10,14)(11,15)(12,13), (1,3,5)(2,4,6)(7,19,12)(8,20,10)(9,21,11)(13,16,23)(14,17,24)(15,18,22), (3,5)(4,6)(7,12)(8,10)(9,11)(13,16)(14,17)(15,18) );
G=PermutationGroup([[(1,22),(2,21),(3,15),(4,11),(5,18),(6,9),(7,8),(10,12),(13,14),(16,17),(19,20),(23,24)], [(1,23),(2,19),(3,13),(4,12),(5,16),(6,7),(8,9),(10,11),(14,15),(17,18),(20,21),(22,24)], [(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,2),(3,4),(10,14),(11,15),(12,13),(19,23),(20,24),(21,22)], [(3,4),(5,6),(7,16),(8,17),(9,18),(10,14),(11,15),(12,13)], [(1,3,5),(2,4,6),(7,19,12),(8,20,10),(9,21,11),(13,16,23),(14,17,24),(15,18,22)], [(3,5),(4,6),(7,12),(8,10),(9,11),(13,16),(14,17),(15,18)]])
G:=TransitiveGroup(24,634);
►On 24 points - transitive group 24T635
Generators in S24
(1 22)(2 21)(3 15)(4 11)(5 18)(6 9)(7 8)(10 12)(13 14)(16 17)(19 20)(23 24)
(1 23)(2 19)(3 13)(4 12)(5 16)(6 7)(8 9)(10 11)(14 15)(17 18)(20 21)(22 24)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 2)(3 4)(10 14)(11 15)(12 13)(19 23)(20 24)(21 22)
(3 4)(5 6)(7 16)(8 17)(9 18)(10 14)(11 15)(12 13)
(1 3 5)(2 4 6)(7 19 12)(8 20 10)(9 21 11)(13 16 23)(14 17 24)(15 18 22)
(1 2)(3 6)(4 5)(7 13)(8 14)(9 15)(10 17)(11 18)(12 16)(19 23)(20 24)(21 22)
G:=sub<Sym(24)| (1,22)(2,21)(3,15)(4,11)(5,18)(6,9)(7,8)(10,12)(13,14)(16,17)(19,20)(23,24), (1,23)(2,19)(3,13)(4,12)(5,16)(6,7)(8,9)(10,11)(14,15)(17,18)(20,21)(22,24), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,4)(10,14)(11,15)(12,13)(19,23)(20,24)(21,22), (3,4)(5,6)(7,16)(8,17)(9,18)(10,14)(11,15)(12,13), (1,3,5)(2,4,6)(7,19,12)(8,20,10)(9,21,11)(13,16,23)(14,17,24)(15,18,22), (1,2)(3,6)(4,5)(7,13)(8,14)(9,15)(10,17)(11,18)(12,16)(19,23)(20,24)(21,22)>;
G:=Group( (1,22)(2,21)(3,15)(4,11)(5,18)(6,9)(7,8)(10,12)(13,14)(16,17)(19,20)(23,24), (1,23)(2,19)(3,13)(4,12)(5,16)(6,7)(8,9)(10,11)(14,15)(17,18)(20,21)(22,24), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,4)(10,14)(11,15)(12,13)(19,23)(20,24)(21,22), (3,4)(5,6)(7,16)(8,17)(9,18)(10,14)(11,15)(12,13), (1,3,5)(2,4,6)(7,19,12)(8,20,10)(9,21,11)(13,16,23)(14,17,24)(15,18,22), (1,2)(3,6)(4,5)(7,13)(8,14)(9,15)(10,17)(11,18)(12,16)(19,23)(20,24)(21,22) );
G=PermutationGroup([[(1,22),(2,21),(3,15),(4,11),(5,18),(6,9),(7,8),(10,12),(13,14),(16,17),(19,20),(23,24)], [(1,23),(2,19),(3,13),(4,12),(5,16),(6,7),(8,9),(10,11),(14,15),(17,18),(20,21),(22,24)], [(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,2),(3,4),(10,14),(11,15),(12,13),(19,23),(20,24),(21,22)], [(3,4),(5,6),(7,16),(8,17),(9,18),(10,14),(11,15),(12,13)], [(1,3,5),(2,4,6),(7,19,12),(8,20,10),(9,21,11),(13,16,23),(14,17,24),(15,18,22)], [(1,2),(3,6),(4,5),(7,13),(8,14),(9,15),(10,17),(11,18),(12,16),(19,23),(20,24),(21,22)]])
G:=TransitiveGroup(24,635);
►On 24 points - transitive group 24T638
Generators in S24
(1 8)(2 9)(4 23)(5 24)(10 20)(12 19)(14 16)(15 17)
(2 9)(3 7)(5 24)(6 22)(10 20)(11 21)(13 18)(15 17)
(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 12)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)
(1 14)(2 15)(3 13)(4 19)(5 20)(6 21)(7 18)(8 16)(9 17)(10 24)(11 22)(12 23)
(4 19 16)(5 20 17)(6 21 18)(10 15 24)(11 13 22)(12 14 23)
(1 8)(2 9)(3 7)(4 12)(5 10)(6 11)(13 18)(14 16)(15 17)(19 23)(20 24)(21 22)
G:=sub<Sym(24)| (1,8)(2,9)(4,23)(5,24)(10,20)(12,19)(14,16)(15,17), (2,9)(3,7)(5,24)(6,22)(10,20)(11,21)(13,18)(15,17), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,14)(2,15)(3,13)(4,19)(5,20)(6,21)(7,18)(8,16)(9,17)(10,24)(11,22)(12,23), (4,19,16)(5,20,17)(6,21,18)(10,15,24)(11,13,22)(12,14,23), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,18)(14,16)(15,17)(19,23)(20,24)(21,22)>;
G:=Group( (1,8)(2,9)(4,23)(5,24)(10,20)(12,19)(14,16)(15,17), (2,9)(3,7)(5,24)(6,22)(10,20)(11,21)(13,18)(15,17), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,14)(2,15)(3,13)(4,19)(5,20)(6,21)(7,18)(8,16)(9,17)(10,24)(11,22)(12,23), (4,19,16)(5,20,17)(6,21,18)(10,15,24)(11,13,22)(12,14,23), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,18)(14,16)(15,17)(19,23)(20,24)(21,22) );
G=PermutationGroup([[(1,8),(2,9),(4,23),(5,24),(10,20),(12,19),(14,16),(15,17)], [(2,9),(3,7),(5,24),(6,22),(10,20),(11,21),(13,18),(15,17)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,12),(2,10),(3,11),(4,16),(5,17),(6,18),(7,21),(8,19),(9,20),(13,22),(14,23),(15,24)], [(1,14),(2,15),(3,13),(4,19),(5,20),(6,21),(7,18),(8,16),(9,17),(10,24),(11,22),(12,23)], [(4,19,16),(5,20,17),(6,21,18),(10,15,24),(11,13,22),(12,14,23)], [(1,8),(2,9),(3,7),(4,12),(5,10),(6,11),(13,18),(14,16),(15,17),(19,23),(20,24),(21,22)]])
G:=TransitiveGroup(24,638);
►On 24 points - transitive group 24T639
Generators in S24
(1 8)(2 9)(4 23)(5 24)(10 20)(12 19)(14 16)(15 17)
(2 9)(3 7)(5 24)(6 22)(10 20)(11 21)(13 18)(15 17)
(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 12)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)
(1 14)(2 15)(3 13)(4 19)(5 20)(6 21)(7 18)(8 16)(9 17)(10 24)(11 22)(12 23)
(4 19 16)(5 20 17)(6 21 18)(10 15 24)(11 13 22)(12 14 23)
(4 19)(5 20)(6 21)(10 24)(11 22)(12 23)
G:=sub<Sym(24)| (1,8)(2,9)(4,23)(5,24)(10,20)(12,19)(14,16)(15,17), (2,9)(3,7)(5,24)(6,22)(10,20)(11,21)(13,18)(15,17), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,14)(2,15)(3,13)(4,19)(5,20)(6,21)(7,18)(8,16)(9,17)(10,24)(11,22)(12,23), (4,19,16)(5,20,17)(6,21,18)(10,15,24)(11,13,22)(12,14,23), (4,19)(5,20)(6,21)(10,24)(11,22)(12,23)>;
G:=Group( (1,8)(2,9)(4,23)(5,24)(10,20)(12,19)(14,16)(15,17), (2,9)(3,7)(5,24)(6,22)(10,20)(11,21)(13,18)(15,17), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,14)(2,15)(3,13)(4,19)(5,20)(6,21)(7,18)(8,16)(9,17)(10,24)(11,22)(12,23), (4,19,16)(5,20,17)(6,21,18)(10,15,24)(11,13,22)(12,14,23), (4,19)(5,20)(6,21)(10,24)(11,22)(12,23) );
G=PermutationGroup([[(1,8),(2,9),(4,23),(5,24),(10,20),(12,19),(14,16),(15,17)], [(2,9),(3,7),(5,24),(6,22),(10,20),(11,21),(13,18),(15,17)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,12),(2,10),(3,11),(4,16),(5,17),(6,18),(7,21),(8,19),(9,20),(13,22),(14,23),(15,24)], [(1,14),(2,15),(3,13),(4,19),(5,20),(6,21),(7,18),(8,16),(9,17),(10,24),(11,22),(12,23)], [(4,19,16),(5,20,17),(6,21,18),(10,15,24),(11,13,22),(12,14,23)], [(4,19),(5,20),(6,21),(10,24),(11,22),(12,23)]])
G:=TransitiveGroup(24,639);
►On 24 points - transitive group 24T705
Generators in S24
(1 10)(2 17)(3 21)(4 8)(5 13)(6 23)(7 14)(9 22)(11 20)(12 18)(15 24)(16 19)
(1 19)(2 11)(3 18)(4 24)(5 9)(6 14)(7 23)(8 15)(10 16)(12 21)(13 22)(17 20)
(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)(2 17)(3 18)(4 24)(5 22)(6 23)(7 14)(8 15)(9 13)(10 19)(11 20)(12 21)
(1 19)(2 20)(3 21)(4 15)(5 13)(6 14)(7 23)(8 24)(9 22)(10 16)(11 17)(12 18)
(1 2 3)(4 14 22)(5 15 23)(6 13 24)(7 9 8)(10 17 21)(11 18 19)(12 16 20)
(1 7)(2 8)(3 9)(4 17)(5 18)(6 16)(10 14)(11 15)(12 13)(19 23)(20 24)(21 22)
G:=sub<Sym(24)| (1,10)(2,17)(3,21)(4,8)(5,13)(6,23)(7,14)(9,22)(11,20)(12,18)(15,24)(16,19), (1,19)(2,11)(3,18)(4,24)(5,9)(6,14)(7,23)(8,15)(10,16)(12,21)(13,22)(17,20), (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)(2,17)(3,18)(4,24)(5,22)(6,23)(7,14)(8,15)(9,13)(10,19)(11,20)(12,21), (1,19)(2,20)(3,21)(4,15)(5,13)(6,14)(7,23)(8,24)(9,22)(10,16)(11,17)(12,18), (1,2,3)(4,14,22)(5,15,23)(6,13,24)(7,9,8)(10,17,21)(11,18,19)(12,16,20), (1,7)(2,8)(3,9)(4,17)(5,18)(6,16)(10,14)(11,15)(12,13)(19,23)(20,24)(21,22)>;
G:=Group( (1,10)(2,17)(3,21)(4,8)(5,13)(6,23)(7,14)(9,22)(11,20)(12,18)(15,24)(16,19), (1,19)(2,11)(3,18)(4,24)(5,9)(6,14)(7,23)(8,15)(10,16)(12,21)(13,22)(17,20), (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)(2,17)(3,18)(4,24)(5,22)(6,23)(7,14)(8,15)(9,13)(10,19)(11,20)(12,21), (1,19)(2,20)(3,21)(4,15)(5,13)(6,14)(7,23)(8,24)(9,22)(10,16)(11,17)(12,18), (1,2,3)(4,14,22)(5,15,23)(6,13,24)(7,9,8)(10,17,21)(11,18,19)(12,16,20), (1,7)(2,8)(3,9)(4,17)(5,18)(6,16)(10,14)(11,15)(12,13)(19,23)(20,24)(21,22) );
G=PermutationGroup([[(1,10),(2,17),(3,21),(4,8),(5,13),(6,23),(7,14),(9,22),(11,20),(12,18),(15,24),(16,19)], [(1,19),(2,11),(3,18),(4,24),(5,9),(6,14),(7,23),(8,15),(10,16),(12,21),(13,22),(17,20)], [(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),(2,17),(3,18),(4,24),(5,22),(6,23),(7,14),(8,15),(9,13),(10,19),(11,20),(12,21)], [(1,19),(2,20),(3,21),(4,15),(5,13),(6,14),(7,23),(8,24),(9,22),(10,16),(11,17),(12,18)], [(1,2,3),(4,14,22),(5,15,23),(6,13,24),(7,9,8),(10,17,21),(11,18,19),(12,16,20)], [(1,7),(2,8),(3,9),(4,17),(5,18),(6,16),(10,14),(11,15),(12,13),(19,23),(20,24),(21,22)]])
G:=TransitiveGroup(24,705);
Matrix representation of A4×S4 ►in GL6(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 1 |
| 0 | 0 | 0 | -1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 1 | -1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | -1 | 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | -1 | 0 | 0 | 0 |
| 1 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,-1,-1,-1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[0,0,1,0,0,0,-1,-1,-1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
A4×S4 in GAP, Magma, Sage, TeX
A_4\times S_4
% in TeX
G:=Group("A4xS4"); // GroupNames label
G:=SmallGroup(288,1024);
// by ID
G=gap.SmallGroup(288,1024);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-3,-2,2,198,94,1684,6053,285,3534,475]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^3=d^2=e^2=f^3=g^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=f^-1>;
// generators/relations
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