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## G = C3×C3⋊S3order 54 = 2·33

### Direct product of C3 and C3⋊S3

Aliases: C3×C3⋊S3, C332C2, C323S3, C324C6, 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 >

Character table of C3×C3⋊S3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 6A 6B size 1 9 1 1 2 2 2 2 2 2 2 2 2 2 2 2 9 9 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 ζ3 ζ32 ζ3 1 ζ3 ζ3 ζ3 1 1 ζ32 ζ32 ζ32 ζ32 1 ζ65 ζ6 linear of order 6 ρ4 1 1 ζ32 ζ3 ζ32 1 ζ32 ζ32 ζ32 1 1 ζ3 ζ3 ζ3 ζ3 1 ζ32 ζ3 linear of order 3 ρ5 1 1 ζ3 ζ32 ζ3 1 ζ3 ζ3 ζ3 1 1 ζ32 ζ32 ζ32 ζ32 1 ζ3 ζ32 linear of order 3 ρ6 1 -1 ζ32 ζ3 ζ32 1 ζ32 ζ32 ζ32 1 1 ζ3 ζ3 ζ3 ζ3 1 ζ6 ζ65 linear of order 6 ρ7 2 0 2 2 -1 -1 -1 2 -1 2 -1 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ8 2 0 2 2 -1 -1 -1 -1 2 -1 2 -1 -1 2 -1 -1 0 0 orthogonal lifted from S3 ρ9 2 0 2 2 -1 2 2 -1 -1 -1 -1 2 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ10 2 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 0 0 orthogonal lifted from S3 ρ11 2 0 -1+√-3 -1-√-3 -1+√-3 -1 ζ65 ζ65 ζ65 -1 -1 ζ6 ζ6 ζ6 -1-√-3 2 0 0 complex lifted from C3×S3 ρ12 2 0 -1+√-3 -1-√-3 ζ65 -1 ζ65 -1+√-3 ζ65 2 -1 ζ6 -1-√-3 ζ6 ζ6 -1 0 0 complex lifted from C3×S3 ρ13 2 0 -1+√-3 -1-√-3 ζ65 2 -1+√-3 ζ65 ζ65 -1 -1 -1-√-3 ζ6 ζ6 ζ6 -1 0 0 complex lifted from C3×S3 ρ14 2 0 -1-√-3 -1+√-3 ζ6 -1 ζ6 -1-√-3 ζ6 2 -1 ζ65 -1+√-3 ζ65 ζ65 -1 0 0 complex lifted from C3×S3 ρ15 2 0 -1-√-3 -1+√-3 ζ6 -1 ζ6 ζ6 -1-√-3 -1 2 ζ65 ζ65 -1+√-3 ζ65 -1 0 0 complex lifted from C3×S3 ρ16 2 0 -1-√-3 -1+√-3 -1-√-3 -1 ζ6 ζ6 ζ6 -1 -1 ζ65 ζ65 ζ65 -1+√-3 2 0 0 complex lifted from C3×S3 ρ17 2 0 -1+√-3 -1-√-3 ζ65 -1 ζ65 ζ65 -1+√-3 -1 2 ζ6 ζ6 -1-√-3 ζ6 -1 0 0 complex lifted from C3×S3 ρ18 2 0 -1-√-3 -1+√-3 ζ6 2 -1-√-3 ζ6 ζ6 -1 -1 -1+√-3 ζ65 ζ65 ζ65 -1 0 0 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  C324D6  C32⋊C18  C32⋊D9  C322D9  C33⋊S3  He34S3  C33.S3  He35S3  C324F7
C3×C3⋊S3 is a maximal quotient of
He34S3  C33.S3  He3.4S3  He3.4C6  C324F7

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

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 0 0 0 4 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 3 0 0 0 6 0 0 0 0 0 1 0 0 1 0
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

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