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## G = A4×S4order 288 = 25·32

### Direct product of A4 and S4

Aliases: A4×S4, A422C2, A4⋊(C2×A4), C22⋊(S3×A4), (C22×S4)⋊C3, (C22×A4)⋊C6, C222(C3×S4), C241(C3×S3), (C22×A4)⋊1S3, SmallGroup(288,1024)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C22×A4 — A4×S4
 Chief series C1 — C22 — A4 — C22×A4 — A42 — A4×S4
 Lower central C22×A4 — A4×S4
 Upper central C1

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 [×5], C3 [×3], C4 [×2], C22 [×2], C22 [×10], S3 [×2], C6 [×3], C2×C4 [×2], D4 [×6], C23 [×7], C32, C12, A4 [×2], A4 [×3], D6 [×2], C2×C6 [×3], C22×C4, C2×D4 [×4], C24, C24, C3×S3, C3×D4, S4, S4, C2×A4 [×3], C22×S3, C22×D4, C3×A4 [×2], C4×A4, C2×S4 [×2], C22×A4 [×2], 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

Permutation representations of A4×S4
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 19)(2 13)(3 23)(4 10)(5 7)(6 17)(8 9)(11 12)(14 15)(16 18)(20 21)(22 24)
(1 20)(2 14)(3 24)(4 11)(5 8)(6 18)(7 9)(10 12)(13 15)(16 17)(19 21)(22 23)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 2)(3 4)(10 23)(11 24)(12 22)(13 19)(14 20)(15 21)
(3 4)(5 6)(7 17)(8 18)(9 16)(10 23)(11 24)(12 22)
(1 3 5)(2 4 6)(7 19 23)(8 20 24)(9 21 22)(10 17 13)(11 18 14)(12 16 15)
(3 5)(4 6)(7 23)(8 24)(9 22)(10 17)(11 18)(12 16)

G:=sub<Sym(24)| (1,19)(2,13)(3,23)(4,10)(5,7)(6,17)(8,9)(11,12)(14,15)(16,18)(20,21)(22,24), (1,20)(2,14)(3,24)(4,11)(5,8)(6,18)(7,9)(10,12)(13,15)(16,17)(19,21)(22,23), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,4)(10,23)(11,24)(12,22)(13,19)(14,20)(15,21), (3,4)(5,6)(7,17)(8,18)(9,16)(10,23)(11,24)(12,22), (1,3,5)(2,4,6)(7,19,23)(8,20,24)(9,21,22)(10,17,13)(11,18,14)(12,16,15), (3,5)(4,6)(7,23)(8,24)(9,22)(10,17)(11,18)(12,16)>;

G:=Group( (1,19)(2,13)(3,23)(4,10)(5,7)(6,17)(8,9)(11,12)(14,15)(16,18)(20,21)(22,24), (1,20)(2,14)(3,24)(4,11)(5,8)(6,18)(7,9)(10,12)(13,15)(16,17)(19,21)(22,23), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,4)(10,23)(11,24)(12,22)(13,19)(14,20)(15,21), (3,4)(5,6)(7,17)(8,18)(9,16)(10,23)(11,24)(12,22), (1,3,5)(2,4,6)(7,19,23)(8,20,24)(9,21,22)(10,17,13)(11,18,14)(12,16,15), (3,5)(4,6)(7,23)(8,24)(9,22)(10,17)(11,18)(12,16) );

G=PermutationGroup([(1,19),(2,13),(3,23),(4,10),(5,7),(6,17),(8,9),(11,12),(14,15),(16,18),(20,21),(22,24)], [(1,20),(2,14),(3,24),(4,11),(5,8),(6,18),(7,9),(10,12),(13,15),(16,17),(19,21),(22,23)], [(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,2),(3,4),(10,23),(11,24),(12,22),(13,19),(14,20),(15,21)], [(3,4),(5,6),(7,17),(8,18),(9,16),(10,23),(11,24),(12,22)], [(1,3,5),(2,4,6),(7,19,23),(8,20,24),(9,21,22),(10,17,13),(11,18,14),(12,16,15)], [(3,5),(4,6),(7,23),(8,24),(9,22),(10,17),(11,18),(12,16)])

G:=TransitiveGroup(24,634);

On 24 points - transitive group 24T635
Generators in S24
(1 19)(2 8)(3 11)(4 18)(5 15)(6 23)(7 9)(10 12)(13 14)(16 17)(20 21)(22 24)
(1 20)(2 9)(3 12)(4 16)(5 13)(6 24)(7 8)(10 11)(14 15)(17 18)(19 21)(22 23)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 2)(3 4)(7 21)(8 19)(9 20)(10 17)(11 18)(12 16)
(3 4)(5 6)(10 17)(11 18)(12 16)(13 24)(14 22)(15 23)
(1 3 5)(2 4 6)(7 17 22)(8 18 23)(9 16 24)(10 14 21)(11 15 19)(12 13 20)
(1 2)(3 6)(4 5)(7 21)(8 19)(9 20)(10 22)(11 23)(12 24)(13 16)(14 17)(15 18)

G:=sub<Sym(24)| (1,19)(2,8)(3,11)(4,18)(5,15)(6,23)(7,9)(10,12)(13,14)(16,17)(20,21)(22,24), (1,20)(2,9)(3,12)(4,16)(5,13)(6,24)(7,8)(10,11)(14,15)(17,18)(19,21)(22,23), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,4)(7,21)(8,19)(9,20)(10,17)(11,18)(12,16), (3,4)(5,6)(10,17)(11,18)(12,16)(13,24)(14,22)(15,23), (1,3,5)(2,4,6)(7,17,22)(8,18,23)(9,16,24)(10,14,21)(11,15,19)(12,13,20), (1,2)(3,6)(4,5)(7,21)(8,19)(9,20)(10,22)(11,23)(12,24)(13,16)(14,17)(15,18)>;

G:=Group( (1,19)(2,8)(3,11)(4,18)(5,15)(6,23)(7,9)(10,12)(13,14)(16,17)(20,21)(22,24), (1,20)(2,9)(3,12)(4,16)(5,13)(6,24)(7,8)(10,11)(14,15)(17,18)(19,21)(22,23), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,4)(7,21)(8,19)(9,20)(10,17)(11,18)(12,16), (3,4)(5,6)(10,17)(11,18)(12,16)(13,24)(14,22)(15,23), (1,3,5)(2,4,6)(7,17,22)(8,18,23)(9,16,24)(10,14,21)(11,15,19)(12,13,20), (1,2)(3,6)(4,5)(7,21)(8,19)(9,20)(10,22)(11,23)(12,24)(13,16)(14,17)(15,18) );

G=PermutationGroup([(1,19),(2,8),(3,11),(4,18),(5,15),(6,23),(7,9),(10,12),(13,14),(16,17),(20,21),(22,24)], [(1,20),(2,9),(3,12),(4,16),(5,13),(6,24),(7,8),(10,11),(14,15),(17,18),(19,21),(22,23)], [(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,2),(3,4),(7,21),(8,19),(9,20),(10,17),(11,18),(12,16)], [(3,4),(5,6),(10,17),(11,18),(12,16),(13,24),(14,22),(15,23)], [(1,3,5),(2,4,6),(7,17,22),(8,18,23),(9,16,24),(10,14,21),(11,15,19),(12,13,20)], [(1,2),(3,6),(4,5),(7,21),(8,19),(9,20),(10,22),(11,23),(12,24),(13,16),(14,17),(15,18)])

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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