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G = C3×C3⋊S3  order 54 = 2·33

Direct product of C3 and C3⋊S3

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

Aliases: C3×C3⋊S3, C33⋊2C2, C32⋊3S3, C32⋊4C6, C3⋊(C3×S3), SmallGroup(54,13)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊S3
Chief series
C1 — C3 — C32 — C33 — C3×C3⋊S3
Lower central
C32 — C3×C3⋊S3
Upper central
C1 — C3

Generators and relations for C3×C3⋊S3
 G = < a,b,c,d | a3=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

C1
9C2
C3
C3
C3
C3
C3
2C3
2C3
2C3
2C3
3S3
3S3
3S3
3S3
9C6
C32
C32
C32
C32
C32
2C32
2C32
2C32
2C32
C3⋊S3
3C3×S3
3C3×S3
3C3×S3
3C3×S3
C33
C3×C3⋊S3

Character table of C3×C3⋊S3

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M3N6A6B
 size 191122222222222299
ρ1111111111111111111    trivial
ρ21-111111111111111-1-1    linear of order 2
ρ31-1ζ3ζ32ζ31ζ3ζ3ζ311ζ32ζ32ζ32ζ321ζ65ζ6    linear of order 6
ρ411ζ32ζ3ζ321ζ32ζ32ζ3211ζ3ζ3ζ3ζ31ζ32ζ3    linear of order 3
ρ511ζ3ζ32ζ31ζ3ζ3ζ311ζ32ζ32ζ32ζ321ζ3ζ32    linear of order 3
ρ61-1ζ32ζ3ζ321ζ32ζ32ζ3211ζ3ζ3ζ3ζ31ζ6ζ65    linear of order 6
ρ72022-1-1-12-12-1-12-1-1-100    orthogonal lifted from S3
ρ82022-1-1-1-12-12-1-12-1-100    orthogonal lifted from S3
ρ92022-122-1-1-1-12-1-1-1-100    orthogonal lifted from S3
ρ1020222-1-1-1-1-1-1-1-1-12200    orthogonal lifted from S3
ρ1120-1+√-3-1-√-3-1+√-3-1ζ65ζ65ζ65-1-1ζ6ζ6ζ6-1-√-3200    complex lifted from C3×S3
ρ1220-1+√-3-1-√-3ζ65-1ζ65-1+√-3ζ652-1ζ6-1-√-3ζ6ζ6-100    complex lifted from C3×S3
ρ1320-1+√-3-1-√-3ζ652-1+√-3ζ65ζ65-1-1-1-√-3ζ6ζ6ζ6-100    complex lifted from C3×S3
ρ1420-1-√-3-1+√-3ζ6-1ζ6-1-√-3ζ62-1ζ65-1+√-3ζ65ζ65-100    complex lifted from C3×S3
ρ1520-1-√-3-1+√-3ζ6-1ζ6ζ6-1-√-3-12ζ65ζ65-1+√-3ζ65-100    complex lifted from C3×S3
ρ1620-1-√-3-1+√-3-1-√-3-1ζ6ζ6ζ6-1-1ζ65ζ65ζ65-1+√-3200    complex lifted from C3×S3
ρ1720-1+√-3-1-√-3ζ65-1ζ65ζ65-1+√-3-12ζ6ζ6-1-√-3ζ6-100    complex lifted from C3×S3
ρ1820-1-√-3-1+√-3ζ62-1-√-3ζ6ζ6-1-1-1+√-3ζ65ζ65ζ65-100    complex lifted from C3×S3

Permutation representations of C3×C3⋊S3
►On 18 points - transitive group 18T23
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)

(1 14 12)(2 15 10)(3 13 11)(4 18 7)(5 16 8)(6 17 9)

(1 10 13)(2 11 14)(3 12 15)(4 9 16)(5 7 17)(6 8 18)

(1 7)(2 8)(3 9)(4 12)(5 10)(6 11)(13 17)(14 18)(15 16)
 
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,14,12)(2,15,10)(3,13,11)(4,18,7)(5,16,8)(6,17,9), (1,10,13)(2,11,14)(3,12,15)(4,9,16)(5,7,17)(6,8,18), (1,7)(2,8)(3,9)(4,12)(5,10)(6,11)(13,17)(14,18)(15,16)>;
 
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,14,12)(2,15,10)(3,13,11)(4,18,7)(5,16,8)(6,17,9), (1,10,13)(2,11,14)(3,12,15)(4,9,16)(5,7,17)(6,8,18), (1,7)(2,8)(3,9)(4,12)(5,10)(6,11)(13,17)(14,18)(15,16) );
 
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,14,12),(2,15,10),(3,13,11),(4,18,7),(5,16,8),(6,17,9)], [(1,10,13),(2,11,14),(3,12,15),(4,9,16),(5,7,17),(6,8,18)], [(1,7),(2,8),(3,9),(4,12),(5,10),(6,11),(13,17),(14,18),(15,16)]])
 
G:=TransitiveGroup(18,23);
 

►On 27 points - transitive group 27T13
Generators in S27
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

(1 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)

(1 14 6)(2 15 4)(3 13 5)(7 16 24)(8 17 22)(9 18 23)(10 19 27)(11 20 25)(12 21 26)

(4 15)(5 13)(6 14)(7 25)(8 26)(9 27)(10 23)(11 24)(12 22)(16 20)(17 21)(18 19)
 
G:=sub<Sym(27)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (4,15)(5,13)(6,14)(7,25)(8,26)(9,27)(10,23)(11,24)(12,22)(16,20)(17,21)(18,19)>;
 
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)(25,26,27), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (4,15)(5,13)(6,14)(7,25)(8,26)(9,27)(10,23)(11,24)(12,22)(16,20)(17,21)(18,19) );
 
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),(25,26,27)], [(1,26,8),(2,27,9),(3,25,7),(4,19,23),(5,20,24),(6,21,22),(10,18,15),(11,16,13),(12,17,14)], [(1,14,6),(2,15,4),(3,13,5),(7,16,24),(8,17,22),(9,18,23),(10,19,27),(11,20,25),(12,21,26)], [(4,15),(5,13),(6,14),(7,25),(8,26),(9,27),(10,23),(11,24),(12,22),(16,20),(17,21),(18,19)]])
 
G:=TransitiveGroup(27,13);
 

C3×C3⋊S3 is a maximal subgroup of
 C33⋊C4  C3×S32  C32⋊4D6  C32⋊C18  C32⋊D9  C32⋊2D9  C33⋊S3  He3⋊4S3  C33.S3  He3⋊5S3  C32⋊4F7
C3×C3⋊S3 is a maximal quotient of
 He3⋊4S3  C33.S3  He3.4S3  He3.4C6  C32⋊4F7

Matrix representation of C3×C3⋊S3 ►in GL4(𝔽7) generated by

2000
0200
0020
0002
,
4000
4200
0010
0001
,
1000
0100
0020
0004
,
1300
0600
0001
0010
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,4,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4],[1,0,0,0,3,6,0,0,0,0,0,1,0,0,1,0] >;
 

C3×C3⋊S3 in GAP, Magma, Sage, TeX 

C_3\times C_3\rtimes S_3
 
% in TeX
 
G:=Group("C3xC3:S3");
 
// GroupNames label
 
G:=SmallGroup(54,13);
 
// by ID
 
G=gap.SmallGroup(54,13);
 
# by ID
 
G:=PCGroup([4,-2,-3,-3,-3,146,579]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C3×C3⋊S3 in TeX
Character table of C3×C3⋊S3 in TeX

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