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## G = C9⋊C12order 108 = 22·33

### The semidirect product of C9 and C12 acting via C12/C2=C6

Aliases: C9⋊C12, C18.C6, Dic9⋊C3, 3- 1+2⋊C4, C32.Dic3, C2.(C9⋊C6), C6.3(C3×S3), (C3×C6).2S3, C3.3(C3×Dic3), (C2×3- 1+2).C2, SmallGroup(108,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C9⋊C12
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C9⋊C12
 Lower central C9 — C9⋊C12
 Upper central C1 — C2

Generators and relations for C9⋊C12
G = < a,b | a9=b12=1, bab-1=a5 >

Character table of C9⋊C12

 class 1 2 3A 3B 3C 4A 4B 6A 6B 6C 9A 9B 9C 12A 12B 12C 12D 18A 18B 18C size 1 1 2 3 3 9 9 2 3 3 6 6 6 9 9 9 9 6 6 6 ρ1 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 linear of order 2 ρ3 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 linear of order 3 ρ4 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 linear of order 3 ρ5 1 1 1 ζ32 ζ3 -1 -1 1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ65 ζ6 ζ32 ζ3 1 linear of order 6 ρ6 1 1 1 ζ3 ζ32 -1 -1 1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ6 ζ65 ζ3 ζ32 1 linear of order 6 ρ7 1 -1 1 1 1 i -i -1 -1 -1 1 1 1 i -i i -i -1 -1 -1 linear of order 4 ρ8 1 -1 1 1 1 -i i -1 -1 -1 1 1 1 -i i -i i -1 -1 -1 linear of order 4 ρ9 1 -1 1 ζ32 ζ3 i -i -1 ζ65 ζ6 ζ3 ζ32 1 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ6 ζ65 -1 linear of order 12 ρ10 1 -1 1 ζ3 ζ32 i -i -1 ζ6 ζ65 ζ32 ζ3 1 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ65 ζ6 -1 linear of order 12 ρ11 1 -1 1 ζ32 ζ3 -i i -1 ζ65 ζ6 ζ3 ζ32 1 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ6 ζ65 -1 linear of order 12 ρ12 1 -1 1 ζ3 ζ32 -i i -1 ζ6 ζ65 ζ32 ζ3 1 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ65 ζ6 -1 linear of order 12 ρ13 2 2 2 2 2 0 0 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ14 2 -2 2 2 2 0 0 -2 -2 -2 -1 -1 -1 0 0 0 0 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 2 -1+√-3 -1-√-3 0 0 2 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 0 0 ζ65 ζ6 -1 complex lifted from C3×S3 ρ16 2 2 2 -1-√-3 -1+√-3 0 0 2 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 0 0 ζ6 ζ65 -1 complex lifted from C3×S3 ρ17 2 -2 2 -1+√-3 -1-√-3 0 0 -2 1+√-3 1-√-3 ζ6 ζ65 -1 0 0 0 0 ζ3 ζ32 1 complex lifted from C3×Dic3 ρ18 2 -2 2 -1-√-3 -1+√-3 0 0 -2 1-√-3 1+√-3 ζ65 ζ6 -1 0 0 0 0 ζ32 ζ3 1 complex lifted from C3×Dic3 ρ19 6 6 -3 0 0 0 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ20 6 -6 -3 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C9⋊C12
On 36 points
Generators in S36
```(1 29 21 11 25 13 5 33 17)(2 14 30 6 22 34 12 18 26)(3 35 15 9 31 19 7 27 23)(4 20 36 8 16 28 10 24 32)
(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 28 29 30 31 32 33 34 35 36)```

`G:=sub<Sym(36)| (1,29,21,11,25,13,5,33,17)(2,14,30,6,22,34,12,18,26)(3,35,15,9,31,19,7,27,23)(4,20,36,8,16,28,10,24,32), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)>;`

`G:=Group( (1,29,21,11,25,13,5,33,17)(2,14,30,6,22,34,12,18,26)(3,35,15,9,31,19,7,27,23)(4,20,36,8,16,28,10,24,32), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36) );`

`G=PermutationGroup([[(1,29,21,11,25,13,5,33,17),(2,14,30,6,22,34,12,18,26),(3,35,15,9,31,19,7,27,23),(4,20,36,8,16,28,10,24,32)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)]])`

C9⋊C12 is a maximal subgroup of
C36.C6  C4×C9⋊C6  Dic9⋊C6  C33⋊Dic3  He3.3Dic3  3- 1+2.Dic3  C33.Dic3  He3.4Dic3  C32.CSU2(𝔽3)  C62.Dic3  Dic9.A4  Dic9⋊A4
C9⋊C12 is a maximal quotient of
C9⋊C24  C32⋊Dic9  C9⋊C36  C33.Dic3  C62.Dic3  Dic9⋊A4

Matrix representation of C9⋊C12 in GL8(𝔽37)

 0 1 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 36 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 1 20 0 0 0 0 0 0 19 36 0 0 0 0 0 0 0 0 33 12 0 0 0 0 0 0 8 4 0 0 0 0 0 0 0 0 0 0 12 29 0 0 0 0 0 0 4 25 0 0 0 0 33 12 0 0 0 0 0 0 8 4 0 0

`G:=sub<GL(8,GF(37))| [0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0],[1,19,0,0,0,0,0,0,20,36,0,0,0,0,0,0,0,0,33,8,0,0,0,0,0,0,12,4,0,0,0,0,0,0,0,0,0,0,33,8,0,0,0,0,0,0,12,4,0,0,0,0,12,4,0,0,0,0,0,0,29,25,0,0] >;`

C9⋊C12 in GAP, Magma, Sage, TeX

`C_9\rtimes C_{12}`
`% in TeX`

`G:=Group("C9:C12");`
`// GroupNames label`

`G:=SmallGroup(108,9);`
`// by ID`

`G=gap.SmallGroup(108,9);`
`# by ID`

`G:=PCGroup([5,-2,-3,-2,-3,-3,30,1203,488,138,1804]);`
`// Polycyclic`

`G:=Group<a,b|a^9=b^12=1,b*a*b^-1=a^5>;`
`// generators/relations`

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