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## G = C6×S4order 144 = 24·32

### Direct product of C6 and S4

Aliases: C6×S4, (C2×A4)⋊C6, A4⋊(C2×C6), (C2×C6)⋊2D6, C22⋊(S3×C6), C23⋊(C3×S3), (C6×A4)⋊1C2, (C22×C6)⋊1S3, (C3×A4)⋊2C22, SmallGroup(144,188)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×S4
G = < a,b,c,d,e | a6=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 >

Subgroups: 216 in 70 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, C12, A4, A4, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, C2×C12, C3×D4, S4, C2×A4, C2×A4, C22×C6, C22×C6, C3×A4, S3×C6, C6×D4, C2×S4, C3×S4, C6×A4, C6×S4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C6×S4

Character table of C6×S4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D size 1 1 3 3 6 6 1 1 8 8 8 6 6 1 1 3 3 3 3 6 6 6 6 8 8 8 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 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 -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 -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 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 ζ3 ζ32 ζ32 1 ζ3 -1 -1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 ζ32 1 ζ3 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ6 1 -1 1 -1 1 -1 ζ3 ζ32 ζ32 1 ζ3 -1 1 ζ6 ζ65 ζ65 ζ6 ζ32 ζ3 ζ6 ζ65 ζ3 ζ32 ζ6 -1 ζ65 ζ6 ζ3 ζ65 ζ32 linear of order 6 ρ7 1 1 1 1 -1 -1 ζ32 ζ3 ζ3 1 ζ32 -1 -1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 ζ3 1 ζ32 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ8 1 -1 1 -1 1 -1 ζ32 ζ3 ζ3 1 ζ32 -1 1 ζ65 ζ6 ζ6 ζ65 ζ3 ζ32 ζ65 ζ6 ζ32 ζ3 ζ65 -1 ζ6 ζ65 ζ32 ζ6 ζ3 linear of order 6 ρ9 1 -1 1 -1 -1 1 ζ32 ζ3 ζ3 1 ζ32 1 -1 ζ65 ζ6 ζ6 ζ65 ζ3 ζ32 ζ3 ζ32 ζ6 ζ65 ζ65 -1 ζ6 ζ3 ζ6 ζ32 ζ65 linear of order 6 ρ10 1 -1 1 -1 -1 1 ζ3 ζ32 ζ32 1 ζ3 1 -1 ζ6 ζ65 ζ65 ζ6 ζ32 ζ3 ζ32 ζ3 ζ65 ζ6 ζ6 -1 ζ65 ζ32 ζ65 ζ3 ζ6 linear of order 6 ρ11 1 1 1 1 1 1 ζ3 ζ32 ζ32 1 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ12 1 1 1 1 1 1 ζ32 ζ3 ζ3 1 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ13 2 -2 2 -2 0 0 2 2 -1 -1 -1 0 0 -2 -2 -2 -2 2 2 0 0 0 0 1 1 1 0 0 0 0 orthogonal lifted from D6 ρ14 2 2 2 2 0 0 2 2 -1 -1 -1 0 0 2 2 2 2 2 2 0 0 0 0 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ15 2 2 2 2 0 0 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 -1+√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 0 0 0 0 ζ65 -1 ζ6 0 0 0 0 complex lifted from C3×S3 ρ16 2 2 2 2 0 0 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 -1-√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 0 0 0 0 ζ6 -1 ζ65 0 0 0 0 complex lifted from C3×S3 ρ17 2 -2 2 -2 0 0 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 1-√-3 1+√-3 1+√-3 1-√-3 -1+√-3 -1-√-3 0 0 0 0 ζ3 1 ζ32 0 0 0 0 complex lifted from S3×C6 ρ18 2 -2 2 -2 0 0 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 1+√-3 1-√-3 1-√-3 1+√-3 -1-√-3 -1+√-3 0 0 0 0 ζ32 1 ζ3 0 0 0 0 complex lifted from S3×C6 ρ19 3 3 -1 -1 1 1 3 3 0 0 0 -1 -1 3 3 -1 -1 -1 -1 1 1 1 1 0 0 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ20 3 3 -1 -1 -1 -1 3 3 0 0 0 1 1 3 3 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 1 1 1 1 orthogonal lifted from S4 ρ21 3 -3 -1 1 -1 1 3 3 0 0 0 -1 1 -3 -3 1 1 -1 -1 1 1 -1 -1 0 0 0 -1 1 -1 1 orthogonal lifted from C2×S4 ρ22 3 -3 -1 1 1 -1 3 3 0 0 0 1 -1 -3 -3 1 1 -1 -1 -1 -1 1 1 0 0 0 1 -1 1 -1 orthogonal lifted from C2×S4 ρ23 3 -3 -1 1 1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 1 -1 3-3√-3/2 3+3√-3/2 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 ζ32 ζ3 0 0 0 ζ3 ζ6 ζ32 ζ65 complex faithful ρ24 3 -3 -1 1 1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 1 -1 3+3√-3/2 3-3√-3/2 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 ζ3 ζ32 0 0 0 ζ32 ζ65 ζ3 ζ6 complex faithful ρ25 3 3 -1 -1 1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 -1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ65 ζ65 ζ6 ζ3 ζ32 ζ32 ζ3 0 0 0 ζ65 ζ6 ζ6 ζ65 complex lifted from C3×S4 ρ26 3 3 -1 -1 1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 -1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ6 ζ6 ζ65 ζ32 ζ3 ζ3 ζ32 0 0 0 ζ6 ζ65 ζ65 ζ6 complex lifted from C3×S4 ρ27 3 -3 -1 1 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 1 3-3√-3/2 3+3√-3/2 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 ζ6 ζ65 0 0 0 ζ65 ζ32 ζ6 ζ3 complex faithful ρ28 3 3 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 1 1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ6 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 0 0 0 ζ32 ζ3 ζ3 ζ32 complex lifted from C3×S4 ρ29 3 -3 -1 1 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 1 3+3√-3/2 3-3√-3/2 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 ζ65 ζ6 0 0 0 ζ6 ζ3 ζ65 ζ32 complex faithful ρ30 3 3 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 1 1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ65 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 0 0 0 ζ3 ζ32 ζ32 ζ3 complex lifted from C3×S4

Permutation representations of C6×S4
On 18 points - transitive group 18T61
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 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 15 9)(2 16 10)(3 17 11)(4 18 12)(5 13 7)(6 14 8)
(1 4)(2 5)(3 6)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)

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

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

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

G:=TransitiveGroup(18,61);

On 24 points - transitive group 24T254
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 9)(3 10)(4 11)(5 12)(6 7)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(7 17 19)(8 18 20)(9 13 21)(10 14 22)(11 15 23)(12 16 24)
(7 19)(8 20)(9 21)(10 22)(11 23)(12 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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (7,17,19)(8,18,20)(9,13,21)(10,14,22)(11,15,23)(12,16,24), (7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;

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

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

G:=TransitiveGroup(24,254);

C6×S4 is a maximal subgroup of   Dic3⋊S4  D6⋊S4

Matrix representation of C6×S4 in GL3(𝔽7) generated by

 3 0 0 0 3 0 0 0 3
,
 0 3 2 5 0 3 0 0 6
,
 2 5 4 4 1 3 3 5 3
,
 2 5 4 0 3 4 0 4 2
,
 2 5 4 3 2 5 6 2 4
G:=sub<GL(3,GF(7))| [3,0,0,0,3,0,0,0,3],[0,5,0,3,0,0,2,3,6],[2,4,3,5,1,5,4,3,3],[2,0,0,5,3,4,4,4,2],[2,3,6,5,2,2,4,5,4] >;

C6×S4 in GAP, Magma, Sage, TeX

C_6\times S_4
% in TeX

G:=Group("C6xS4");
// GroupNames label

G:=SmallGroup(144,188);
// by ID

G=gap.SmallGroup(144,188);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,2,579,2164,202,1301,347]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=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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