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## G = C33⋊C12order 324 = 22·34

### 1st semidirect product of C33 and C12 acting via C12/C2=C6

Aliases: C331C12, He31Dic3, C3≀C33C4, C335C41C3, (C2×He3).1S3, (C32×C6).2C6, C2.(C33⋊C6), C6.2(C32⋊C6), C3.2(C32⋊C12), C32.6(C3×Dic3), (C2×C3≀C3).3C2, (C3×C6).13(C3×S3), SmallGroup(324,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊C12
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C2×C3≀C3 — C33⋊C12
 Lower central C33 — C33⋊C12
 Upper central C1 — C2

Generators and relations for C33⋊C12
G = < a,b,c,d | a3=b3=c3=d12=1, ab=ba, ac=ca, dad-1=a-1bc-1, bc=cb, dbd-1=b-1c, dcd-1=c-1 >

Character table of C33⋊C12

 class 1 2 3A 3B 3C 3D 3E 3F 3G 4A 4B 6A 6B 6C 6D 6E 6F 6G 9A 9B 12A 12B 12C 12D 18A 18B size 1 1 2 6 6 6 6 9 9 27 27 2 6 6 6 6 9 9 18 18 27 27 27 27 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 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 linear of order 2 ρ3 1 1 1 1 1 1 1 ζ3 ζ32 -1 -1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ4 1 1 1 1 1 1 1 ζ32 ζ3 -1 -1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ5 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ6 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ7 1 -1 1 1 1 1 1 1 1 -i i -1 -1 -1 -1 -1 -1 -1 1 1 -i i -i i -1 -1 linear of order 4 ρ8 1 -1 1 1 1 1 1 1 1 i -i -1 -1 -1 -1 -1 -1 -1 1 1 i -i i -i -1 -1 linear of order 4 ρ9 1 -1 1 1 1 1 1 ζ3 ζ32 -i i -1 -1 -1 -1 -1 ζ6 ζ65 ζ3 ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ6 ζ65 linear of order 12 ρ10 1 -1 1 1 1 1 1 ζ3 ζ32 i -i -1 -1 -1 -1 -1 ζ6 ζ65 ζ3 ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ6 ζ65 linear of order 12 ρ11 1 -1 1 1 1 1 1 ζ32 ζ3 i -i -1 -1 -1 -1 -1 ζ65 ζ6 ζ32 ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ65 ζ6 linear of order 12 ρ12 1 -1 1 1 1 1 1 ζ32 ζ3 -i i -1 -1 -1 -1 -1 ζ65 ζ6 ζ32 ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ65 ζ6 linear of order 12 ρ13 2 2 2 -1 -1 2 -1 2 2 0 0 2 -1 2 -1 -1 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ14 2 -2 2 -1 -1 2 -1 2 2 0 0 -2 1 -2 1 1 -2 -2 -1 -1 0 0 0 0 1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 2 -1 -1 2 -1 -1+√-3 -1-√-3 0 0 2 -1 2 -1 -1 -1-√-3 -1+√-3 ζ65 ζ6 0 0 0 0 ζ6 ζ65 complex lifted from C3×S3 ρ16 2 -2 2 -1 -1 2 -1 -1+√-3 -1-√-3 0 0 -2 1 -2 1 1 1+√-3 1-√-3 ζ65 ζ6 0 0 0 0 ζ32 ζ3 complex lifted from C3×Dic3 ρ17 2 -2 2 -1 -1 2 -1 -1-√-3 -1+√-3 0 0 -2 1 -2 1 1 1-√-3 1+√-3 ζ6 ζ65 0 0 0 0 ζ3 ζ32 complex lifted from C3×Dic3 ρ18 2 2 2 -1 -1 2 -1 -1-√-3 -1+√-3 0 0 2 -1 2 -1 -1 -1+√-3 -1-√-3 ζ6 ζ65 0 0 0 0 ζ65 ζ6 complex lifted from C3×S3 ρ19 6 6 -3 -3 0 0 3 0 0 0 0 -3 0 0 3 -3 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊C6 ρ20 6 6 -3 0 3 0 -3 0 0 0 0 -3 3 0 -3 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊C6 ρ21 6 6 -3 3 -3 0 0 0 0 0 0 -3 -3 0 0 3 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊C6 ρ22 6 6 6 0 0 -3 0 0 0 0 0 6 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ23 6 -6 -3 -3 0 0 3 0 0 0 0 3 0 0 -3 3 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ24 6 -6 -3 0 3 0 -3 0 0 0 0 3 -3 0 3 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ25 6 -6 6 0 0 -3 0 0 0 0 0 -6 0 3 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊C12, Schur index 2 ρ26 6 -6 -3 3 -3 0 0 0 0 0 0 3 3 0 0 -3 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C33⋊C12
On 36 points
Generators in S36
```(2 33 19)(5 22 36)(8 27 13)(11 16 30)
(2 19 33)(3 20 34)(5 36 22)(6 25 23)(8 13 27)(9 14 28)(11 30 16)(12 31 17)
(1 18 32)(2 33 19)(3 20 34)(4 35 21)(5 22 36)(6 25 23)(7 24 26)(8 27 13)(9 14 28)(10 29 15)(11 16 30)(12 31 17)
(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)| (2,33,19)(5,22,36)(8,27,13)(11,16,30), (2,19,33)(3,20,34)(5,36,22)(6,25,23)(8,13,27)(9,14,28)(11,30,16)(12,31,17), (1,18,32)(2,33,19)(3,20,34)(4,35,21)(5,22,36)(6,25,23)(7,24,26)(8,27,13)(9,14,28)(10,29,15)(11,16,30)(12,31,17), (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( (2,33,19)(5,22,36)(8,27,13)(11,16,30), (2,19,33)(3,20,34)(5,36,22)(6,25,23)(8,13,27)(9,14,28)(11,30,16)(12,31,17), (1,18,32)(2,33,19)(3,20,34)(4,35,21)(5,22,36)(6,25,23)(7,24,26)(8,27,13)(9,14,28)(10,29,15)(11,16,30)(12,31,17), (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([[(2,33,19),(5,22,36),(8,27,13),(11,16,30)], [(2,19,33),(3,20,34),(5,36,22),(6,25,23),(8,13,27),(9,14,28),(11,30,16),(12,31,17)], [(1,18,32),(2,33,19),(3,20,34),(4,35,21),(5,22,36),(6,25,23),(7,24,26),(8,27,13),(9,14,28),(10,29,15),(11,16,30),(12,31,17)], [(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)]])`

Matrix representation of C33⋊C12 in GL8(𝔽37)

 36 36 0 0 0 0 0 0 1 0 0 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36
,
 1 0 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 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36
,
 6 0 0 0 0 0 0 0 31 31 0 0 0 0 0 0 0 0 0 0 5 10 0 0 0 0 0 0 5 32 0 0 0 0 0 0 0 0 5 10 0 0 0 0 0 0 5 32 0 0 5 10 0 0 0 0 0 0 5 32 0 0 0 0

`G:=sub<GL(8,GF(37))| [36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,36,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36],[6,31,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,10,32,0,0,5,5,0,0,0,0,0,0,10,32,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,10,32,0,0] >;`

C33⋊C12 in GAP, Magma, Sage, TeX

`C_3^3\rtimes C_{12}`
`% in TeX`

`G:=Group("C3^3:C12");`
`// GroupNames label`

`G:=SmallGroup(324,14);`
`// by ID`

`G=gap.SmallGroup(324,14);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,579,585,2164,2170,7781]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b*c^-1,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=c^-1>;`
`// generators/relations`

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