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## G = C3×S4order 72 = 23·32

### Direct product of C3 and S4

Aliases: C3×S4, A4⋊C6, (C2×C6)⋊1S3, C22⋊(C3×S3), (C3×A4)⋊1C2, SmallGroup(72,42)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×S4
G = < a,b,c,d,e | a3=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Character table of C3×S4

 class 1 2A 2B 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 12A 12B size 1 3 6 1 1 8 8 8 6 3 3 6 6 6 6 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 ζ3 ζ32 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 -1 ζ32 ζ3 ζ3 ζ32 1 -1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ5 1 1 -1 ζ3 ζ32 ζ32 ζ3 1 -1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ6 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ7 2 2 0 2 2 -1 -1 -1 0 2 2 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 0 -1-√-3 -1+√-3 0 0 0 0 complex lifted from C3×S3 ρ9 2 2 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 0 -1+√-3 -1-√-3 0 0 0 0 complex lifted from C3×S3 ρ10 3 -1 -1 3 3 0 0 0 1 -1 -1 -1 -1 1 1 orthogonal lifted from S4 ρ11 3 -1 1 3 3 0 0 0 -1 -1 -1 1 1 -1 -1 orthogonal lifted from S4 ρ12 3 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 ζ6 ζ65 ζ3 ζ32 ζ65 ζ6 complex faithful ρ13 3 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 ζ65 ζ6 ζ32 ζ3 ζ6 ζ65 complex faithful ρ14 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 1 ζ6 ζ65 ζ65 ζ6 ζ3 ζ32 complex faithful ρ15 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 1 ζ65 ζ6 ζ6 ζ65 ζ32 ζ3 complex faithful

Permutation representations of C3×S4
On 12 points - transitive group 12T45
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 4)(2 5)(3 6)(7 12)(8 10)(9 11)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)
(4 8 10)(5 9 11)(6 7 12)
(4 10)(5 11)(6 12)

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

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

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

G:=TransitiveGroup(12,45);

On 18 points - transitive group 18T30
Generators in S18
(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)| (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( (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([[(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(18,30);

On 18 points - transitive group 18T33
Generators in S18
(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)| (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( (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([[(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(18,33);

On 24 points - transitive group 24T80
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 15)(2 13)(3 14)(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)(10 18)(11 16)(12 17)
(1 18)(2 16)(3 17)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(19 24)(20 22)(21 23)
(7 19 24)(8 20 22)(9 21 23)(10 15 18)(11 13 16)(12 14 17)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)

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

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

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

G:=TransitiveGroup(24,80);

On 24 points - transitive group 24T84
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 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)| (1,2,3)(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( (1,2,3)(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([[(1,2,3),(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(24,84);

C3×S4 is a maximal subgroup of   C32⋊S4
C3×S4 is a maximal quotient of   C32.S4  C62⋊S3

Polynomial with Galois group C3×S4 over ℚ
actionf(x)Disc(f)
12T45x12-4x9-13x8-11x6-5x5+26x4+x3+x2-2x-1-78·2833·18196512

Matrix representation of C3×S4 in GL3(𝔽7) generated by

 4 0 0 0 4 0 0 0 4
,
 6 0 0 0 6 0 0 0 1
,
 1 0 0 0 6 0 0 0 6
,
 0 0 3 5 0 0 0 1 0
,
 6 0 0 0 0 6 0 6 0
G:=sub<GL(3,GF(7))| [4,0,0,0,4,0,0,0,4],[6,0,0,0,6,0,0,0,1],[1,0,0,0,6,0,0,0,6],[0,5,0,0,0,1,3,0,0],[6,0,0,0,0,6,0,6,0] >;

C3×S4 in GAP, Magma, Sage, TeX

C_3\times S_4
% in TeX

G:=Group("C3xS4");
// GroupNames label

G:=SmallGroup(72,42);
// by ID

G=gap.SmallGroup(72,42);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,2,182,723,133,454,239]);
// Polycyclic

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

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