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G = A4×S4order 288 = 25·32

Direct product of A4 and S4

direct product, non-abelian, soluble, monomial

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

C1C22C22×A4 — A4×S4
C1C22A4C22×A4A42 — A4×S4
C22×A4 — A4×S4
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 12A2B2C2D2E3A3B3C3D3E4A4B6A6B6C6D6E12A12B
 size 1336918448323261812122424242424
ρ111111111111111111111    trivial
ρ2111-11-111111-1-1111-1-1-1-1    linear of order 2
ρ3111-11-1ζ3ζ321ζ32ζ3-1-1ζ3ζ321ζ6ζ65ζ6ζ65    linear of order 6
ρ4111111ζ3ζ321ζ32ζ311ζ3ζ321ζ32ζ3ζ32ζ3    linear of order 3
ρ5111-11-1ζ32ζ31ζ3ζ32-1-1ζ32ζ31ζ65ζ6ζ65ζ6    linear of order 6
ρ6111111ζ32ζ31ζ3ζ3211ζ32ζ31ζ3ζ32ζ3ζ32    linear of order 3
ρ722202022-1-1-10022-10000    orthogonal lifted from S3
ρ8222020-1+-3-1--3-1ζ6ζ6500-1+-3-1--3-10000    complex lifted from C3×S3
ρ9222020-1--3-1+-3-1ζ65ζ600-1--3-1+-3-10000    complex lifted from C3×S3
ρ103-13-3-1100300-3100-10000    orthogonal lifted from C2×A4
ρ113-133-1-1003003-100-10000    orthogonal lifted from A4
ρ1233-11-1133000-1-1-1-1011-1-1    orthogonal lifted from S4
ρ1333-1-1-1-13300011-1-10-1-111    orthogonal lifted from S4
ρ1433-11-11-3-3-3/2-3+3-3/2000-1-1ζ6ζ650ζ3ζ32ζ65ζ6    complex lifted from C3×S4
ρ1533-1-1-1-1-3-3-3/2-3+3-3/200011ζ6ζ650ζ65ζ6ζ3ζ32    complex lifted from C3×S4
ρ1633-11-11-3+3-3/2-3-3-3/2000-1-1ζ65ζ60ζ32ζ3ζ6ζ65    complex lifted from C3×S4
ρ1733-1-1-1-1-3+3-3/2-3-3-3/200011ζ65ζ60ζ6ζ65ζ32ζ3    complex lifted from C3×S4
ρ186-260-2000-300000010000    orthogonal lifted from S3×A4
ρ199-3-331-100000-310000000    orthogonal faithful
ρ209-3-3-311000003-10000000    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)

100000
010000
001000
000-100
000-101
000-110
,
100000
010000
001000
0000-11
0000-10
0001-10
,
100000
010000
001000
000001
000100
000010
,
0-11000
0-10000
1-10000
000100
000010
000001
,
01-1000
10-1000
00-1000
000100
000010
000001
,
001000
100000
010000
000100
000010
000001
,
010000
100000
001000
000100
000010
000001

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

Export

Character table of A4×S4 in TeX

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