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## G = C9⋊C18order 162 = 2·34

### The semidirect product of C9 and C18 acting via C18/C3=C6

Aliases: C9⋊C18, D9⋊C9, C9⋊C9⋊C2, (C3×C9).C6, (C3×D9).C3, C3.3(S3×C9), (C3×C9).1S3, C3.2(C9⋊C6), C32.14(C3×S3), SmallGroup(162,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C9⋊C18
 Chief series C1 — C3 — C9 — C3×C9 — C9⋊C9 — C9⋊C18
 Lower central C9 — C9⋊C18
 Upper central C1 — C3

Generators and relations for C9⋊C18
G = < a,b | a9=b18=1, bab-1=a2 >

Character table of C9⋊C18

 class 1 2 3A 3B 3C 3D 3E 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 9M 9N 9O 18A 18B 18C 18D 18E 18F size 1 9 1 1 2 2 2 9 9 3 3 3 3 3 3 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 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 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 ζ3 ζ32 ζ3 1 ζ3 ζ32 1 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ4 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 ζ32 ζ3 ζ32 1 ζ32 ζ3 1 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ5 1 -1 1 1 1 1 1 -1 -1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 ζ32 ζ3 ζ32 1 ζ32 ζ3 1 ζ3 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 linear of order 6 ρ6 1 -1 1 1 1 1 1 -1 -1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 ζ3 ζ32 ζ3 1 ζ3 ζ32 1 ζ32 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 linear of order 6 ρ7 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ94 ζ9 ζ97 ζ98 ζ95 ζ92 1 ζ94 ζ92 ζ9 ζ32 ζ97 ζ98 ζ3 ζ95 ζ92 ζ98 ζ95 ζ94 ζ9 ζ97 linear of order 9 ρ8 1 -1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ94 ζ9 ζ97 ζ98 ζ95 ζ92 1 ζ94 ζ92 ζ9 ζ32 ζ97 ζ98 ζ3 ζ95 -ζ92 -ζ98 -ζ95 -ζ94 -ζ9 -ζ97 linear of order 18 ρ9 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ9 ζ97 ζ94 ζ92 ζ98 ζ95 1 ζ9 ζ95 ζ97 ζ32 ζ94 ζ92 ζ3 ζ98 ζ95 ζ92 ζ98 ζ9 ζ97 ζ94 linear of order 9 ρ10 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ92 ζ95 ζ98 ζ94 ζ97 ζ9 1 ζ92 ζ9 ζ95 ζ3 ζ98 ζ94 ζ32 ζ97 ζ9 ζ94 ζ97 ζ92 ζ95 ζ98 linear of order 9 ρ11 1 -1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ92 ζ95 ζ98 ζ94 ζ97 ζ9 1 ζ92 ζ9 ζ95 ζ3 ζ98 ζ94 ζ32 ζ97 -ζ9 -ζ94 -ζ97 -ζ92 -ζ95 -ζ98 linear of order 18 ρ12 1 -1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ95 ζ98 ζ92 ζ9 ζ94 ζ97 1 ζ95 ζ97 ζ98 ζ3 ζ92 ζ9 ζ32 ζ94 -ζ97 -ζ9 -ζ94 -ζ95 -ζ98 -ζ92 linear of order 18 ρ13 1 -1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ9 ζ97 ζ94 ζ92 ζ98 ζ95 1 ζ9 ζ95 ζ97 ζ32 ζ94 ζ92 ζ3 ζ98 -ζ95 -ζ92 -ζ98 -ζ9 -ζ97 -ζ94 linear of order 18 ρ14 1 -1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ98 ζ92 ζ95 ζ97 ζ9 ζ94 1 ζ98 ζ94 ζ92 ζ3 ζ95 ζ97 ζ32 ζ9 -ζ94 -ζ97 -ζ9 -ζ98 -ζ92 -ζ95 linear of order 18 ρ15 1 -1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ97 ζ94 ζ9 ζ95 ζ92 ζ98 1 ζ97 ζ98 ζ94 ζ32 ζ9 ζ95 ζ3 ζ92 -ζ98 -ζ95 -ζ92 -ζ97 -ζ94 -ζ9 linear of order 18 ρ16 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ97 ζ94 ζ9 ζ95 ζ92 ζ98 1 ζ97 ζ98 ζ94 ζ32 ζ9 ζ95 ζ3 ζ92 ζ98 ζ95 ζ92 ζ97 ζ94 ζ9 linear of order 9 ρ17 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ95 ζ98 ζ92 ζ9 ζ94 ζ97 1 ζ95 ζ97 ζ98 ζ3 ζ92 ζ9 ζ32 ζ94 ζ97 ζ9 ζ94 ζ95 ζ98 ζ92 linear of order 9 ρ18 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ98 ζ92 ζ95 ζ97 ζ9 ζ94 1 ζ98 ζ94 ζ92 ζ3 ζ95 ζ97 ζ32 ζ9 ζ94 ζ97 ζ9 ζ98 ζ92 ζ95 linear of order 9 ρ19 2 0 2 2 2 2 2 0 0 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S3 ρ20 2 0 2 2 2 2 2 0 0 -1+√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1 ζ65 ζ6 ζ65 -1 ζ65 ζ6 -1 ζ6 0 0 0 0 0 0 complex lifted from C3×S3 ρ21 2 0 2 2 2 2 2 0 0 -1-√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1 ζ6 ζ65 ζ6 -1 ζ6 ζ65 -1 ζ65 0 0 0 0 0 0 complex lifted from C3×S3 ρ22 2 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 0 0 2ζ95 2ζ98 2ζ92 2ζ9 2ζ94 2ζ97 -1 -ζ95 -ζ97 -ζ98 ζ65 -ζ92 -ζ9 ζ6 -ζ94 0 0 0 0 0 0 complex lifted from S3×C9 ρ23 2 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 0 0 2ζ97 2ζ94 2ζ9 2ζ95 2ζ92 2ζ98 -1 -ζ97 -ζ98 -ζ94 ζ6 -ζ9 -ζ95 ζ65 -ζ92 0 0 0 0 0 0 complex lifted from S3×C9 ρ24 2 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 0 0 2ζ98 2ζ92 2ζ95 2ζ97 2ζ9 2ζ94 -1 -ζ98 -ζ94 -ζ92 ζ65 -ζ95 -ζ97 ζ6 -ζ9 0 0 0 0 0 0 complex lifted from S3×C9 ρ25 2 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 0 0 2ζ92 2ζ95 2ζ98 2ζ94 2ζ97 2ζ9 -1 -ζ92 -ζ9 -ζ95 ζ65 -ζ98 -ζ94 ζ6 -ζ97 0 0 0 0 0 0 complex lifted from S3×C9 ρ26 2 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 0 0 2ζ94 2ζ9 2ζ97 2ζ98 2ζ95 2ζ92 -1 -ζ94 -ζ92 -ζ9 ζ6 -ζ97 -ζ98 ζ65 -ζ95 0 0 0 0 0 0 complex lifted from S3×C9 ρ27 2 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 0 0 2ζ9 2ζ97 2ζ94 2ζ92 2ζ98 2ζ95 -1 -ζ9 -ζ95 -ζ97 ζ6 -ζ94 -ζ92 ζ65 -ζ98 0 0 0 0 0 0 complex lifted from S3×C9 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 C9⋊C18
On 18 points - transitive group 18T80
Generators in S18
```(1 5 15 13 17 9 7 11 3)(2 16 18 8 4 6 14 10 12)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)```

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

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

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

`G:=TransitiveGroup(18,80);`

C9⋊C18 is a maximal subgroup of
C27⋊C18  C9⋊C9.S3  C9⋊C9.3S3  C9⋊C9⋊S3  C9×C9⋊C6  D9⋊3- 1+2  C927C6  C928C6  C9⋊(S3×C9)  C923S3  C9⋊C92S3  C926S3  C925S3
C9⋊C18 is a maximal quotient of
C9⋊C36  C9⋊S3⋊C9  C9⋊C54  C27⋊C18  C9⋊(S3×C9)

Matrix representation of C9⋊C18 in GL6(𝔽19)

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

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

C9⋊C18 in GAP, Magma, Sage, TeX

`C_9\rtimes C_{18}`
`% in TeX`

`G:=Group("C9:C18");`
`// GroupNames label`

`G:=SmallGroup(162,6);`
`// by ID`

`G=gap.SmallGroup(162,6);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-3,-3,36,1803,728,138,2704]);`
`// Polycyclic`

`G:=Group<a,b|a^9=b^18=1,b*a*b^-1=a^2>;`
`// generators/relations`

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