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G = C3×S3order 18 = 2·32

Direct product of C3 and S3

direct product, metacyclic, supersoluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C3 — C3×S3
C1C3C32 — C3×S3
C3 — C3×S3
C1C3

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

3C2
2C3
3C6

Character table of C3×S3

 class 123A3B3C3D3E6A6B
 size 131122233
ρ1111111111    trivial
ρ21-111111-1-1    linear of order 2
ρ31-1ζ32ζ31ζ32ζ3ζ65ζ6    linear of order 6
ρ41-1ζ3ζ321ζ3ζ32ζ6ζ65    linear of order 6
ρ511ζ3ζ321ζ3ζ32ζ32ζ3    linear of order 3
ρ611ζ32ζ31ζ32ζ3ζ3ζ32    linear of order 3
ρ72022-1-1-100    orthogonal lifted from S3
ρ820-1+-3-1--3-1ζ65ζ600    complex faithful
ρ920-1--3-1+-3-1ζ6ζ6500    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);

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

20
02
,
20
04
,
01
10
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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