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G = C3⋊S3order 18 = 2·32

The semidirect product of C3 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, A-group, rational

Aliases: C3⋊S3, C322C2, SmallGroup(18,4)

Series: Derived Chief Lower central Upper central

C1C32 — C3⋊S3
C1C3C32 — C3⋊S3
C32 — C3⋊S3
C1

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

9C2
3S3
3S3
3S3
3S3

Character table of C3⋊S3

 class 123A3B3C3D
 size 192222
ρ1111111    trivial
ρ21-11111    linear of order 2
ρ320-1-1-12    orthogonal lifted from S3
ρ4202-1-1-1    orthogonal lifted from S3
ρ520-12-1-1    orthogonal lifted from S3
ρ620-1-12-1    orthogonal lifted from S3

Permutation representations of C3⋊S3
On 9 points - transitive group 9T5
Generators in S9
(1 2 3)(4 5 6)(7 8 9)
(1 5 8)(2 6 9)(3 4 7)
(2 3)(4 9)(5 8)(6 7)

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

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

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

G:=TransitiveGroup(9,5);

Regular action on 18 points - transitive group 18T4
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 8)(2 6 9)(3 4 7)(10 16 13)(11 17 14)(12 18 15)
(1 10)(2 12)(3 11)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)

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

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

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

G:=TransitiveGroup(18,4);

Polynomial with Galois group C3⋊S3 over ℚ
actionf(x)Disc(f)
9T5x9-54x7-12x6+756x5+180x4-2652x3-864x2+288x+64232·318·72·238·432·1512

Matrix representation of C3⋊S3 in GL4(ℤ) generated by

0100
-1-100
0001
00-1-1
,
1000
0100
0001
00-1-1
,
1000
-1-100
000-1
00-10
G:=sub<GL(4,Integers())| [0,-1,0,0,1,-1,0,0,0,0,0,-1,0,0,1,-1],[1,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,-1],[1,-1,0,0,0,-1,0,0,0,0,0,-1,0,0,-1,0] >;

C3⋊S3 in GAP, Magma, Sage, TeX

C_3\rtimes S_3
% in TeX

G:=Group("C3:S3");
// GroupNames label

G:=SmallGroup(18,4);
// by ID

G=gap.SmallGroup(18,4);
# by ID

G:=PCGroup([3,-2,-3,-3,25,110]);
// Polycyclic

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

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