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## G = S3×S4order 144 = 24·32

### Direct product of S3 and S4

Aliases: S3×S4, A41D6, C3⋊S4⋊C2, C22⋊S32, (C2×C6)⋊D6, (C3×S4)⋊C2, (S3×A4)⋊C2, C31(C2×S4), (C22×S3)⋊S3, (C3×A4)⋊C22, Hol(C2×C6), SmallGroup(144,183)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — S3×S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — S3×A4 — S3×S4
 Lower central C3×A4 — S3×S4
 Upper central C1

Generators and relations for S3×S4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 372 in 70 conjugacy classes, 13 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, S3, C6, C2×C4, D4, C23, C32, Dic3, C12, A4, A4, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C4×S3, D12, C3⋊D4, C3×D4, S4, S4, C2×A4, C22×S3, C22×S3, S32, C3×A4, S3×D4, C2×S4, C3×S4, C3⋊S4, S3×A4, S3×S4
Quotients: C1, C2, C22, S3, D6, S4, S32, C2×S4, S3×S4

Character table of S3×S4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 6A 6B 6C 12 size 1 3 3 6 9 18 2 8 16 6 18 6 12 24 12 ρ1 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 linear of order 2 ρ3 1 1 1 -1 1 -1 1 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ4 1 1 -1 1 -1 -1 1 1 1 1 -1 1 1 -1 1 linear of order 2 ρ5 2 2 -2 0 -2 0 2 -1 -1 0 0 2 0 1 0 orthogonal lifted from D6 ρ6 2 2 0 -2 0 0 -1 2 -1 -2 0 -1 1 0 1 orthogonal lifted from D6 ρ7 2 2 0 2 0 0 -1 2 -1 2 0 -1 -1 0 -1 orthogonal lifted from S3 ρ8 2 2 2 0 2 0 2 -1 -1 0 0 2 0 -1 0 orthogonal lifted from S3 ρ9 3 -1 3 -1 -1 -1 3 0 0 1 1 -1 -1 0 1 orthogonal lifted from S4 ρ10 3 -1 -3 1 1 -1 3 0 0 -1 1 -1 1 0 -1 orthogonal lifted from C2×S4 ρ11 3 -1 3 1 -1 1 3 0 0 -1 -1 -1 1 0 -1 orthogonal lifted from S4 ρ12 3 -1 -3 -1 1 1 3 0 0 1 -1 -1 -1 0 1 orthogonal lifted from C2×S4 ρ13 4 4 0 0 0 0 -2 -2 1 0 0 -2 0 0 0 orthogonal lifted from S32 ρ14 6 -2 0 -2 0 0 -3 0 0 2 0 1 1 0 -1 orthogonal faithful ρ15 6 -2 0 2 0 0 -3 0 0 -2 0 1 -1 0 1 orthogonal faithful

Permutation representations of S3×S4
On 12 points - transitive group 12T83
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(2 3)(4 5)(8 9)(10 12)
(1 11)(2 12)(3 10)(4 8)(5 9)(6 7)
(1 6)(2 4)(3 5)(7 11)(8 12)(9 10)
(4 8 12)(5 9 10)(6 7 11)
(7 11)(8 12)(9 10)

G:=sub<Sym(12)| (1,2,3)(4,5,6)(7,8,9)(10,11,12), (2,3)(4,5)(8,9)(10,12), (1,11)(2,12)(3,10)(4,8)(5,9)(6,7), (1,6)(2,4)(3,5)(7,11)(8,12)(9,10), (4,8,12)(5,9,10)(6,7,11), (7,11)(8,12)(9,10)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12), (2,3)(4,5)(8,9)(10,12), (1,11)(2,12)(3,10)(4,8)(5,9)(6,7), (1,6)(2,4)(3,5)(7,11)(8,12)(9,10), (4,8,12)(5,9,10)(6,7,11), (7,11)(8,12)(9,10) );

G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(2,3),(4,5),(8,9),(10,12)], [(1,11),(2,12),(3,10),(4,8),(5,9),(6,7)], [(1,6),(2,4),(3,5),(7,11),(8,12),(9,10)], [(4,8,12),(5,9,10),(6,7,11)], [(7,11),(8,12),(9,10)]])

G:=TransitiveGroup(12,83);

On 18 points - transitive group 18T65
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 9)(2 8)(3 7)(4 16)(5 18)(6 17)(10 13)(11 15)(12 14)
(1 9)(2 7)(3 8)(10 14)(11 15)(12 13)
(4 18)(5 16)(6 17)(10 14)(11 15)(12 13)
(1 11 17)(2 12 18)(3 10 16)(4 7 13)(5 8 14)(6 9 15)
(4 13)(5 14)(6 15)(10 16)(11 17)(12 18)

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

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

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

G:=TransitiveGroup(18,65);

On 18 points - transitive group 18T69
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 9)(2 8)(3 7)(4 16)(5 18)(6 17)(10 13)(11 15)(12 14)
(1 9)(2 7)(3 8)(10 14)(11 15)(12 13)
(4 18)(5 16)(6 17)(10 14)(11 15)(12 13)
(1 11 17)(2 12 18)(3 10 16)(4 7 13)(5 8 14)(6 9 15)
(1 9)(2 7)(3 8)(4 12)(5 10)(6 11)(13 18)(14 16)(15 17)

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

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

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

G:=TransitiveGroup(18,69);

On 18 points - transitive group 18T70
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(2 3)(4 5)(7 8)(10 12)(13 14)(16 18)
(1 9)(2 7)(3 8)(10 14)(11 15)(12 13)
(4 18)(5 16)(6 17)(10 14)(11 15)(12 13)
(1 11 17)(2 12 18)(3 10 16)(4 7 13)(5 8 14)(6 9 15)
(1 9)(2 7)(3 8)(4 12)(5 10)(6 11)(13 18)(14 16)(15 17)

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

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

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

G:=TransitiveGroup(18,70);

On 18 points - transitive group 18T72
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(2 3)(4 5)(7 8)(10 12)(13 14)(16 18)
(1 9)(2 7)(3 8)(10 14)(11 15)(12 13)
(4 18)(5 16)(6 17)(10 14)(11 15)(12 13)
(1 11 17)(2 12 18)(3 10 16)(4 7 13)(5 8 14)(6 9 15)
(4 13)(5 14)(6 15)(10 16)(11 17)(12 18)

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

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

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

G:=TransitiveGroup(18,72);

On 24 points - transitive group 24T275
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 7)(3 9)(4 17)(5 16)(6 18)(10 21)(11 20)(12 19)(13 23)(14 22)(15 24)
(1 11)(2 12)(3 10)(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)(13 16)(14 17)(15 18)
(1 17)(2 18)(3 16)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(19 24)(20 22)(21 23)
(4 22 20)(5 23 21)(6 24 19)(10 16 13)(11 17 14)(12 18 15)
(10 13)(11 14)(12 15)(19 24)(20 22)(21 23)

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

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

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

G:=TransitiveGroup(24,275);

On 24 points - transitive group 24T276
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)
(2 3)(4 5)(8 9)(10 12)(13 15)(17 18)(19 20)(22 24)
(1 14)(2 15)(3 13)(4 17)(5 18)(6 16)(7 23)(8 24)(9 22)(10 20)(11 21)(12 19)
(1 21)(2 19)(3 20)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)
(4 17 24)(5 18 22)(6 16 23)(10 13 20)(11 14 21)(12 15 19)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)

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

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

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

G:=TransitiveGroup(24,276);

On 24 points - transitive group 24T277
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 7)(2 9)(3 8)(4 20)(5 19)(6 21)(10 17)(11 16)(12 18)(13 24)(14 23)(15 22)
(1 14)(2 15)(3 13)(4 17)(5 18)(6 16)(7 23)(8 24)(9 22)(10 20)(11 21)(12 19)
(1 21)(2 19)(3 20)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)
(4 17 24)(5 18 22)(6 16 23)(10 13 20)(11 14 21)(12 15 19)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)

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

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

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

G:=TransitiveGroup(24,277);

On 24 points - transitive group 24T281
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 7)(2 9)(3 8)(4 20)(5 19)(6 21)(10 17)(11 16)(12 18)(13 24)(14 23)(15 22)
(1 14)(2 15)(3 13)(4 17)(5 18)(6 16)(7 23)(8 24)(9 22)(10 20)(11 21)(12 19)
(1 21)(2 19)(3 20)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)
(1 3 2)(4 18 23)(5 16 24)(6 17 22)(7 8 9)(10 15 21)(11 13 19)(12 14 20)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)

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

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

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

G:=TransitiveGroup(24,281);

S3×S4 is a maximal subgroup of
C625D6  C6210D6
S3×S4 is a maximal quotient of
CSU2(𝔽3)⋊S3  Dic3.4S4  Dic3.5S4  GL2(𝔽3)⋊S3  D6.S4  D6.2S4  Dic3.S4  Dic32S4  Dic3⋊S4  D6⋊S4  A4⋊D12  C625D6  C6210D6

Polynomial with Galois group S3×S4 over ℚ
actionf(x)Disc(f)
12T83x12-3x9+3x6+x3-3-323·413

Matrix representation of S3×S4 in GL5(ℤ)

 -1 -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 -1 -1 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 1 0 0 0 -1 0 0 0 1 -1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 -1 0 0 1 0 -1 0 0 0 0 -1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(5,Integers())| [-1,1,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,-1,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,-1,-1,-1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,-1,-1,-1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

S3×S4 in GAP, Magma, Sage, TeX

S_3\times S_4
% in TeX

G:=Group("S3xS4");
// GroupNames label

G:=SmallGroup(144,183);
// by ID

G=gap.SmallGroup(144,183);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,2,80,579,1090,556,659,989]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^3=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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