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## G = S3×C9⋊C6order 324 = 22·34

### Direct product of S3 and C9⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — S3×C9⋊C6
 Chief series C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — S3×3- 1+2 — S3×C9⋊C6
 Lower central C3×C9 — S3×C9⋊C6
 Upper central C1

Generators and relations for S3×C9⋊C6
G = < a,b,c,d | a3=b2=c9=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 424 in 80 conjugacy classes, 23 normal (all characteristic)
C1, C2 [×3], C3 [×2], C3 [×3], C22, S3, S3 [×4], C6 [×6], C9, C9 [×3], C32 [×2], C32 [×3], D6 [×2], C2×C6, D9, D9 [×2], C18 [×2], C3×S3, C3×S3 [×7], C3⋊S3, C3×C6 [×2], C3×C9, C3×C9, 3- 1+2, 3- 1+2 [×3], C33, D18, S32, S3×C6 [×2], C3×D9, S3×C9, S3×C9, C9⋊C6, C9⋊C6 [×3], C9⋊S3, C2×3- 1+2, S3×C32, S3×C32, C3×C3⋊S3, C3×3- 1+2, S3×D9, C2×C9⋊C6, C3×S32, C3×C9⋊C6, S3×3- 1+2, C33.S3, S3×C9⋊C6
Quotients: C1, C2 [×3], C3, C22, S3 [×2], C6 [×3], D6 [×2], C2×C6, C3×S3 [×2], S32, S3×C6 [×2], C9⋊C6, C2×C9⋊C6, C3×S32, S3×C9⋊C6

Character table of S3×C9⋊C6

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 9A 9B 9C 9D 9E 9F 18A 18B 18C size 1 3 9 27 2 2 3 3 4 6 6 6 9 9 9 9 18 18 18 27 27 6 6 6 12 12 12 18 18 18 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 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 1 1 1 linear of order 2 ρ3 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 -1 -1 -1 linear of order 2 ρ4 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 -1 -1 -1 linear of order 2 ρ5 1 -1 1 -1 1 1 ζ32 ζ3 1 ζ3 ζ32 -1 ζ65 ζ3 ζ6 ζ32 1 ζ3 ζ32 ζ65 ζ6 1 ζ3 ζ32 ζ3 1 ζ32 ζ6 -1 ζ65 linear of order 6 ρ6 1 -1 -1 1 1 1 ζ32 ζ3 1 ζ3 ζ32 -1 ζ65 ζ65 ζ6 ζ6 -1 ζ65 ζ6 ζ3 ζ32 1 ζ3 ζ32 ζ3 1 ζ32 ζ6 -1 ζ65 linear of order 6 ρ7 1 1 1 1 1 1 ζ32 ζ3 1 ζ3 ζ32 1 ζ3 ζ3 ζ32 ζ32 1 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 ζ3 1 ζ32 ζ32 1 ζ3 linear of order 3 ρ8 1 1 1 1 1 1 ζ3 ζ32 1 ζ32 ζ3 1 ζ32 ζ32 ζ3 ζ3 1 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 ζ32 1 ζ3 ζ3 1 ζ32 linear of order 3 ρ9 1 1 -1 -1 1 1 ζ3 ζ32 1 ζ32 ζ3 1 ζ32 ζ6 ζ3 ζ65 -1 ζ6 ζ65 ζ6 ζ65 1 ζ32 ζ3 ζ32 1 ζ3 ζ3 1 ζ32 linear of order 6 ρ10 1 -1 1 -1 1 1 ζ3 ζ32 1 ζ32 ζ3 -1 ζ6 ζ32 ζ65 ζ3 1 ζ32 ζ3 ζ6 ζ65 1 ζ32 ζ3 ζ32 1 ζ3 ζ65 -1 ζ6 linear of order 6 ρ11 1 -1 -1 1 1 1 ζ3 ζ32 1 ζ32 ζ3 -1 ζ6 ζ6 ζ65 ζ65 -1 ζ6 ζ65 ζ32 ζ3 1 ζ32 ζ3 ζ32 1 ζ3 ζ65 -1 ζ6 linear of order 6 ρ12 1 1 -1 -1 1 1 ζ32 ζ3 1 ζ3 ζ32 1 ζ3 ζ65 ζ32 ζ6 -1 ζ65 ζ6 ζ65 ζ6 1 ζ3 ζ32 ζ3 1 ζ32 ζ32 1 ζ3 linear of order 6 ρ13 2 0 2 0 -1 2 2 2 -1 -1 -1 0 0 2 0 2 -1 -1 -1 0 0 2 2 2 -1 -1 -1 0 0 0 orthogonal lifted from S3 ρ14 2 -2 0 0 2 2 2 2 2 2 2 -2 -2 0 -2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ15 2 0 -2 0 -1 2 2 2 -1 -1 -1 0 0 -2 0 -2 1 1 1 0 0 2 2 2 -1 -1 -1 0 0 0 orthogonal lifted from D6 ρ16 2 2 0 0 2 2 2 2 2 2 2 2 2 0 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ17 2 0 -2 0 -1 2 -1+√-3 -1-√-3 -1 ζ6 ζ65 0 0 1+√-3 0 1-√-3 1 ζ32 ζ3 0 0 2 -1-√-3 -1+√-3 ζ6 -1 ζ65 0 0 0 complex lifted from S3×C6 ρ18 2 0 2 0 -1 2 -1-√-3 -1+√-3 -1 ζ65 ζ6 0 0 -1+√-3 0 -1-√-3 -1 ζ65 ζ6 0 0 2 -1+√-3 -1-√-3 ζ65 -1 ζ6 0 0 0 complex lifted from C3×S3 ρ19 2 2 0 0 2 2 -1+√-3 -1-√-3 2 -1-√-3 -1+√-3 2 -1-√-3 0 -1+√-3 0 0 0 0 0 0 -1 ζ6 ζ65 ζ6 -1 ζ65 ζ65 -1 ζ6 complex lifted from C3×S3 ρ20 2 0 -2 0 -1 2 -1-√-3 -1+√-3 -1 ζ65 ζ6 0 0 1-√-3 0 1+√-3 1 ζ3 ζ32 0 0 2 -1+√-3 -1-√-3 ζ65 -1 ζ6 0 0 0 complex lifted from S3×C6 ρ21 2 -2 0 0 2 2 -1+√-3 -1-√-3 2 -1-√-3 -1+√-3 -2 1+√-3 0 1-√-3 0 0 0 0 0 0 -1 ζ6 ζ65 ζ6 -1 ζ65 ζ3 1 ζ32 complex lifted from S3×C6 ρ22 2 -2 0 0 2 2 -1-√-3 -1+√-3 2 -1+√-3 -1-√-3 -2 1-√-3 0 1+√-3 0 0 0 0 0 0 -1 ζ65 ζ6 ζ65 -1 ζ6 ζ32 1 ζ3 complex lifted from S3×C6 ρ23 2 0 2 0 -1 2 -1+√-3 -1-√-3 -1 ζ6 ζ65 0 0 -1-√-3 0 -1+√-3 -1 ζ6 ζ65 0 0 2 -1-√-3 -1+√-3 ζ6 -1 ζ65 0 0 0 complex lifted from C3×S3 ρ24 2 2 0 0 2 2 -1-√-3 -1+√-3 2 -1+√-3 -1-√-3 2 -1+√-3 0 -1-√-3 0 0 0 0 0 0 -1 ζ65 ζ6 ζ65 -1 ζ6 ζ6 -1 ζ65 complex lifted from C3×S3 ρ25 4 0 0 0 -2 4 4 4 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -2 -2 -2 1 1 1 0 0 0 orthogonal lifted from S32 ρ26 4 0 0 0 -2 4 -2-2√-3 -2+2√-3 -2 1-√-3 1+√-3 0 0 0 0 0 0 0 0 0 0 -2 1-√-3 1+√-3 ζ3 1 ζ32 0 0 0 complex lifted from C3×S32 ρ27 4 0 0 0 -2 4 -2+2√-3 -2-2√-3 -2 1+√-3 1-√-3 0 0 0 0 0 0 0 0 0 0 -2 1+√-3 1-√-3 ζ32 1 ζ3 0 0 0 complex lifted from C3×S32 ρ28 6 -6 0 0 6 -3 0 0 -3 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C9⋊C6 ρ29 6 6 0 0 6 -3 0 0 -3 0 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ30 12 0 0 0 -6 -6 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of S3×C9⋊C6
On 18 points - transitive group 18T122
Generators in S18
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)
(1 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 16)(2 12 8 15 5 18)(3 17 6 14 9 11)(4 13)(7 10)

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

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

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

G:=TransitiveGroup(18,122);

On 27 points - transitive group 27T116
Generators in S27
(1 16 21)(2 17 22)(3 18 23)(4 10 24)(5 11 25)(6 12 26)(7 13 27)(8 14 19)(9 15 20)
(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)
(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)
(2 6 8 9 5 3)(4 7)(10 13)(11 18 17 12 14 15)(19 20 25 23 22 26)(24 27)

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

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

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

G:=TransitiveGroup(27,116);

Matrix representation of S3×C9⋊C6 in GL10(𝔽19)

 18 1 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 18 0 1 0 0 0 0 0 0 0 0 18 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 18 0 18 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 18 0 0 18 0
,
 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 18 0 18 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 0 18 0 0 0 0 0 18 0 0 18 0

G:=sub<GL(10,GF(19))| [18,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[18,0,18,0,0,0,0,0,0,0,0,18,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,1,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,1,0,0,0,0,0],[0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,1,0,0,18,0] >;

S3×C9⋊C6 in GAP, Magma, Sage, TeX

S_3\times C_9\rtimes C_6
% in TeX

G:=Group("S3xC9:C6");
// GroupNames label

G:=SmallGroup(324,118);
// by ID

G=gap.SmallGroup(324,118);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1593,735,453,2164,3899]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^9=d^6=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^2>;
// generators/relations

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