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## G = C3×S3order 18 = 2·32

### Direct product of C3 and S3

Aliases: C3×S3, C3C2, U2(𝔽2), AΣL1(𝔽9), CU2(𝔽2), C3⋊C6, C321C2, SmallGroup(18,3)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×S3
G = < a,b,c | a3=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of C3×S3

 class 1 2 3A 3B 3C 3D 3E 6A 6B size 1 3 1 1 2 2 2 3 3 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 -1 -1 linear of order 2 ρ3 1 -1 ζ32 ζ3 1 ζ32 ζ3 ζ65 ζ6 linear of order 6 ρ4 1 -1 ζ3 ζ32 1 ζ3 ζ32 ζ6 ζ65 linear of order 6 ρ5 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ7 2 0 2 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ8 2 0 -1+√-3 -1-√-3 -1 ζ65 ζ6 0 0 complex faithful ρ9 2 0 -1-√-3 -1+√-3 -1 ζ6 ζ65 0 0 complex faithful

Permutation representations of C3×S3
On 6 points - transitive group 6T5
Generators in S6
(1 2 3)(4 5 6)
(1 2 3)(4 6 5)
(1 4)(2 5)(3 6)

G:=sub<Sym(6)| (1,2,3)(4,5,6), (1,2,3)(4,6,5), (1,4)(2,5)(3,6)>;

G:=Group( (1,2,3)(4,5,6), (1,2,3)(4,6,5), (1,4)(2,5)(3,6) );

G=PermutationGroup([[(1,2,3),(4,5,6)], [(1,2,3),(4,6,5)], [(1,4),(2,5),(3,6)]])

G:=TransitiveGroup(6,5);

On 9 points - transitive group 9T4
Generators in S9
(1 2 3)(4 5 6)(7 8 9)
(1 5 9)(2 6 7)(3 4 8)
(4 8)(5 9)(6 7)

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

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

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

G:=TransitiveGroup(9,4);

Regular action on 18 points - transitive group 18T3
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 10 14)(2 11 15)(3 12 13)(4 18 8)(5 16 9)(6 17 7)
(1 16)(2 17)(3 18)(4 12)(5 10)(6 11)(7 15)(8 13)(9 14)

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

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

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

G:=TransitiveGroup(18,3);

C3×S3 is a maximal subgroup of
C32⋊C6  C9⋊C6  He3⋊C2  C3⋊F7  D39⋊C3  A4≀C2  D57⋊C3  C5⋊D15⋊C3
C3×S3 is a maximal quotient of
C32⋊C6  C9⋊C6  C3⋊F7  D39⋊C3  A4≀C2  D57⋊C3  C5⋊D15⋊C3

Polynomial with Galois group C3×S3 over ℚ
actionf(x)Disc(f)
6T5x6-2x5+2x3-x2+1-26·132
9T4x9-3x8-8x7+13x6+22x5-13x4-20x3+x2+5x+176·533

Matrix representation of C3×S3 in GL2(𝔽7) generated by

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

C3×S3 in GAP, Magma, Sage, TeX

C_3\times S_3
% in TeX

G:=Group("C3xS3");
// GroupNames label

G:=SmallGroup(18,3);
// by ID

G=gap.SmallGroup(18,3);
# by ID

G:=PCGroup([3,-2,-3,-3,110]);
// Polycyclic

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

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