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## G = C3×C9⋊C6order 162 = 2·34

### Direct product of C3 and C9⋊C6

Aliases: C3×C9⋊C6, D9⋊C32, C33.3S3, 3- 1+23C6, C9⋊(C3×C6), (C3×D9)⋊C3, (C3×C9)⋊4C6, C3.3(S3×C32), C32.4(C3×S3), (C3×3- 1+2)⋊1C2, SmallGroup(162,36)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C9⋊C6
G = < a,b,c | a3=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >

Character table of C3×C9⋊C6

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

Permutation representations of C3×C9⋊C6
On 18 points - transitive group 18T83
Generators in S18
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 13 7 16 4 10)(2 18 5 15 8 12)(3 14)(6 11)(9 17)

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

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

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

G:=TransitiveGroup(18,83);

On 27 points - transitive group 27T57
Generators in S27
(1 20 11)(2 21 12)(3 22 13)(4 23 14)(5 24 15)(6 25 16)(7 26 17)(8 27 18)(9 19 10)
(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 11 20)(2 16 27 9 15 22)(3 12 25 8 10 24)(4 17 23 7 14 26)(5 13 21 6 18 19)

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

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

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

G:=TransitiveGroup(27,57);

C3×C9⋊C6 is a maximal subgroup of
D9⋊He3  C927C6  C928C6  C34.7S3  (C32×C9)⋊S3  C33⋊(C3×S3)  He3.C32C6  He3⋊(C3×S3)  C3.He3⋊C6  3- 1+4⋊C2
C3×C9⋊C6 is a maximal quotient of
C34.S3  D9⋊He3  D9⋊3- 1+2  C927C6  C928C6

Matrix representation of C3×C9⋊C6 in GL6(𝔽19)

 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 0 7 0 0 0 0 0 0 7 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 11 0 0 0 0 0 0 11 0
,
 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 7 11 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0] >;

C3×C9⋊C6 in GAP, Magma, Sage, TeX

C_3\times C_9\rtimes C_6
% in TeX

G:=Group("C3xC9:C6");
// GroupNames label

G:=SmallGroup(162,36);
// by ID

G=gap.SmallGroup(162,36);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,1803,728,138,2704]);
// Polycyclic

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

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