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## G = S3×A4order 72 = 23·32

### Direct product of S3 and A4

Aliases: S3×A4, C3⋊(C2×A4), (C2×C6)⋊C6, (C3×A4)⋊3C2, (C22×S3)⋊C3, C222(C3×S3), SmallGroup(72,44)

Series: Derived Chief Lower central Upper central

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

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

Permutation representations of S3×A4
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

Polynomial with Galois group S3×A4 over ℚ
actionf(x)Disc(f)
12T43x12-6x11+18x10-28x9+18x8+36x5+32x3+24x2+4234·324·965892

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

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