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## G = S3×C6order 36 = 22·32

### Direct product of C6 and S3

Aliases: S3×C6, C6⋊C6, C322C22, C3⋊(C2×C6), (C3×C6)⋊1C2, SmallGroup(36,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C6
 Chief series C1 — C3 — C32 — C3×S3 — S3×C6
 Lower central C3 — S3×C6
 Upper central C1 — C6

Generators and relations for S3×C6
G = < a,b,c | a6=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of S3×C6

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I size 1 1 3 3 1 1 2 2 2 1 1 2 2 2 3 3 3 3 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 ζ32 ζ3 ζ3 ζ32 1 ζ6 ζ65 ζ65 -1 ζ6 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ6 1 1 -1 -1 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 ζ32 1 ζ3 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ7 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ8 1 -1 -1 1 ζ32 ζ3 ζ3 ζ32 1 ζ6 ζ65 ζ65 -1 ζ6 ζ32 ζ3 ζ65 ζ6 linear of order 6 ρ9 1 1 -1 -1 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 ζ3 1 ζ32 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ10 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ11 1 -1 -1 1 ζ3 ζ32 ζ32 ζ3 1 ζ65 ζ6 ζ6 -1 ζ65 ζ3 ζ32 ζ6 ζ65 linear of order 6 ρ12 1 -1 1 -1 ζ3 ζ32 ζ32 ζ3 1 ζ65 ζ6 ζ6 -1 ζ65 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ13 2 2 0 0 2 2 -1 -1 -1 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ14 2 -2 0 0 2 2 -1 -1 -1 -2 -2 1 1 1 0 0 0 0 orthogonal lifted from D6 ρ15 2 -2 0 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 1+√-3 1-√-3 ζ3 1 ζ32 0 0 0 0 complex faithful ρ16 2 2 0 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 0 0 complex lifted from C3×S3 ρ17 2 -2 0 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 1-√-3 1+√-3 ζ32 1 ζ3 0 0 0 0 complex faithful ρ18 2 2 0 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 0 0 complex lifted from C3×S3

Permutation representations of S3×C6
On 12 points - transitive group 12T18
Generators in S12
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)

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

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

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

G:=TransitiveGroup(12,18);

On 18 points - transitive group 18T6
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 11 14)(2 12 15)(3 7 16)(4 8 17)(5 9 18)(6 10 13)
(1 4)(2 5)(3 6)(7 13)(8 14)(9 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), (1,11,14)(2,12,15)(3,7,16)(4,8,17)(5,9,18)(6,10,13), (1,4)(2,5)(3,6)(7,13)(8,14)(9,15)(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), (1,11,14)(2,12,15)(3,7,16)(4,8,17)(5,9,18)(6,10,13), (1,4)(2,5)(3,6)(7,13)(8,14)(9,15)(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)], [(1,11,14),(2,12,15),(3,7,16),(4,8,17),(5,9,18),(6,10,13)], [(1,4),(2,5),(3,6),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)])

G:=TransitiveGroup(18,6);

S3×C6 is a maximal subgroup of   D6⋊S3  C3⋊D12

Polynomial with Galois group S3×C6 over ℚ
actionf(x)Disc(f)
12T18x12-2x11+6x9+x8-8x7+3x6+2x4-6x3+7x2-4x+1212·38·56·5092

Matrix representation of S3×C6 in GL2(𝔽7) generated by

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

S3×C6 in GAP, Magma, Sage, TeX

S_3\times C_6
% in TeX

G:=Group("S3xC6");
// GroupNames label

G:=SmallGroup(36,12);
// by ID

G=gap.SmallGroup(36,12);
# by ID

G:=PCGroup([4,-2,-2,-3,-3,387]);
// Polycyclic

G:=Group<a,b,c|a^6=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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