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## G = S3×C32order 54 = 2·33

### Direct product of C32 and S3

Aliases: S3×C32, C331C2, C323C6, C3⋊(C3×C6), SmallGroup(54,12)

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C32
G = < a,b,c,d | a3=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of S3×C32

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 6A 6B 6C 6D 6E 6F 6G 6H size 1 3 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 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 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 ζ3 ζ3 ζ32 1 ζ32 ζ3 1 ζ32 ζ32 1 ζ3 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 linear of order 3 ρ4 1 -1 ζ32 ζ32 ζ3 1 ζ3 ζ32 1 ζ3 ζ3 1 ζ32 ζ32 1 1 ζ3 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 -1 ζ65 -1 ζ6 linear of order 6 ρ5 1 -1 ζ3 1 ζ32 ζ3 1 ζ32 ζ32 ζ3 1 ζ3 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 -1 ζ6 ζ65 -1 ζ6 ζ65 linear of order 6 ρ6 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 ζ32 ζ3 1 ζ3 1 ζ32 ζ3 ζ32 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ7 1 -1 1 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 1 ζ32 ζ65 ζ6 ζ6 -1 ζ65 ζ65 ζ6 -1 linear of order 6 ρ8 1 -1 ζ32 1 ζ3 ζ32 1 ζ3 ζ3 ζ32 1 ζ32 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 -1 ζ65 ζ6 -1 ζ65 ζ6 linear of order 6 ρ9 1 1 ζ32 1 ζ3 ζ32 1 ζ3 ζ3 ζ32 1 ζ32 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 linear of order 3 ρ10 1 1 ζ32 ζ32 ζ3 1 ζ3 ζ32 1 ζ3 ζ3 1 ζ32 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 linear of order 3 ρ11 1 -1 1 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 1 ζ3 ζ6 ζ65 ζ65 -1 ζ6 ζ6 ζ65 -1 linear of order 6 ρ12 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 ζ3 ζ32 1 ζ32 1 ζ3 ζ32 ζ3 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ13 1 -1 ζ3 ζ3 ζ32 1 ζ32 ζ3 1 ζ32 ζ32 1 ζ3 ζ3 1 1 ζ32 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 -1 ζ6 -1 ζ65 linear of order 6 ρ14 1 1 1 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 1 ζ3 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 linear of order 3 ρ15 1 1 ζ3 1 ζ32 ζ3 1 ζ32 ζ32 ζ3 1 ζ3 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 linear of order 3 ρ16 1 -1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 ζ32 ζ3 1 ζ3 1 ζ32 ζ3 ζ32 1 -1 -1 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 linear of order 6 ρ17 1 1 1 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 1 ζ32 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 linear of order 3 ρ18 1 -1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 ζ3 ζ32 1 ζ32 1 ζ3 ζ32 ζ3 1 -1 -1 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 linear of order 6 ρ19 2 0 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ20 2 0 2 -1+√-3 2 -1-√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 ζ6 ζ6 ζ6 ζ65 -1 ζ65 -1 -1 ζ65 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ21 2 0 -1-√-3 -1-√-3 -1+√-3 2 -1+√-3 -1-√-3 2 -1+√-3 ζ65 -1 ζ6 ζ6 -1 -1 ζ65 ζ6 ζ65 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ22 2 0 -1+√-3 -1+√-3 -1-√-3 2 -1-√-3 -1+√-3 2 -1-√-3 ζ6 -1 ζ65 ζ65 -1 -1 ζ6 ζ65 ζ6 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ23 2 0 -1-√-3 2 -1+√-3 -1-√-3 2 -1+√-3 -1+√-3 -1-√-3 -1 ζ6 ζ65 -1 -1 ζ65 ζ65 ζ6 ζ6 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ24 2 0 2 -1-√-3 2 -1+√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 ζ65 ζ65 ζ65 ζ6 -1 ζ6 -1 -1 ζ6 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ25 2 0 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 2 -1-√-3 2 ζ6 ζ65 -1 ζ65 -1 ζ6 ζ65 ζ6 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ26 2 0 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 2 -1+√-3 2 ζ65 ζ6 -1 ζ6 -1 ζ65 ζ6 ζ65 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ27 2 0 -1+√-3 2 -1-√-3 -1+√-3 2 -1-√-3 -1-√-3 -1+√-3 -1 ζ65 ζ6 -1 -1 ζ6 ζ6 ζ65 ζ65 0 0 0 0 0 0 0 0 complex lifted from C3×S3

Permutation representations of S3×C32
On 18 points - transitive group 18T17
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 8 18)(5 9 16)(6 7 17)
(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)(4,8,18)(5,9,16)(6,7,17), (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)(4,8,18)(5,9,16)(6,7,17), (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),(4,8,18),(5,9,16),(6,7,17)], [(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(18,17);

On 27 points - transitive group 27T15
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 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 27 19)(11 25 20)(12 26 21)
(1 8 26)(2 9 27)(3 7 25)(4 23 19)(5 24 20)(6 22 21)(10 15 18)(11 13 16)(12 14 17)
(7 25)(8 26)(9 27)(10 18)(11 16)(12 17)(19 23)(20 24)(21 22)

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

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

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

G:=TransitiveGroup(27,15);

S3×C32 is a maximal subgroup of   C3≀S3

Matrix representation of S3×C32 in GL3(𝔽7) generated by

 2 0 0 0 2 0 0 0 2
,
 4 0 0 0 1 0 0 0 1
,
 1 0 0 0 2 0 0 0 4
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(7))| [2,0,0,0,2,0,0,0,2],[4,0,0,0,1,0,0,0,1],[1,0,0,0,2,0,0,0,4],[1,0,0,0,0,1,0,1,0] >;

S3×C32 in GAP, Magma, Sage, TeX

S_3\times C_3^2
% in TeX

G:=Group("S3xC3^2");
// GroupNames label

G:=SmallGroup(54,12);
// by ID

G=gap.SmallGroup(54,12);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,579]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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