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## G = C32.S4order 216 = 23·33

### The non-split extension by C32 of S4 acting via S4/C22=S3

Aliases: C32.S4, C62.5S3, C3.S4⋊C3, C3.A4⋊C6, C22⋊(C9⋊C6), C3.1(C3×S4), C32.A4⋊C2, (C2×C6).2(C3×S3), SmallGroup(216,90)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3.A4 — C32.S4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C32.A4 — C32.S4
 Lower central C3.A4 — C32.S4
 Upper central C1

Generators and relations for C32.S4
G = < a,b,c,d,e,f | a3=b3=c2=d2=f2=1, e3=b, ab=ba, ac=ca, ad=da, eae-1=ab-1, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=b-1e2 >

Character table of C32.S4

 class 1 2A 2B 3A 3B 3C 4 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12A 12B size 1 3 18 2 3 3 18 3 3 6 6 6 18 18 24 24 24 18 18 ρ1 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 linear of order 2 ρ3 1 1 -1 1 ζ3 ζ32 -1 ζ3 ζ32 ζ3 ζ32 1 ζ65 ζ6 ζ3 1 ζ32 ζ6 ζ65 linear of order 6 ρ4 1 1 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 ζ3 1 ζ32 ζ32 ζ3 linear of order 3 ρ5 1 1 -1 1 ζ32 ζ3 -1 ζ32 ζ3 ζ32 ζ3 1 ζ6 ζ65 ζ32 1 ζ3 ζ65 ζ6 linear of order 6 ρ6 1 1 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 ζ32 1 ζ3 ζ3 ζ32 linear of order 3 ρ7 2 2 0 2 2 2 0 2 2 2 2 2 0 0 -1 -1 -1 0 0 orthogonal lifted from S3 ρ8 2 2 0 2 -1+√-3 -1-√-3 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 0 0 ζ65 -1 ζ6 0 0 complex lifted from C3×S3 ρ9 2 2 0 2 -1-√-3 -1+√-3 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 0 0 ζ6 -1 ζ65 0 0 complex lifted from C3×S3 ρ10 3 -1 1 3 3 3 -1 -1 -1 -1 -1 -1 1 1 0 0 0 -1 -1 orthogonal lifted from S4 ρ11 3 -1 -1 3 3 3 1 -1 -1 -1 -1 -1 -1 -1 0 0 0 1 1 orthogonal lifted from S4 ρ12 3 -1 1 3 -3-3√-3/2 -3+3√-3/2 -1 ζ6 ζ65 ζ6 ζ65 -1 ζ32 ζ3 0 0 0 ζ65 ζ6 complex lifted from C3×S4 ρ13 3 -1 -1 3 -3+3√-3/2 -3-3√-3/2 1 ζ65 ζ6 ζ65 ζ6 -1 ζ65 ζ6 0 0 0 ζ32 ζ3 complex lifted from C3×S4 ρ14 3 -1 1 3 -3+3√-3/2 -3-3√-3/2 -1 ζ65 ζ6 ζ65 ζ6 -1 ζ3 ζ32 0 0 0 ζ6 ζ65 complex lifted from C3×S4 ρ15 3 -1 -1 3 -3-3√-3/2 -3+3√-3/2 1 ζ6 ζ65 ζ6 ζ65 -1 ζ6 ζ65 0 0 0 ζ3 ζ32 complex lifted from C3×S4 ρ16 6 -2 0 -3 0 0 0 4 4 -2 -2 1 0 0 0 0 0 0 0 orthogonal faithful ρ17 6 6 0 -3 0 0 0 0 0 0 0 -3 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ18 6 -2 0 -3 0 0 0 -2-2√-3 -2+2√-3 1+√-3 1-√-3 1 0 0 0 0 0 0 0 complex faithful ρ19 6 -2 0 -3 0 0 0 -2+2√-3 -2-2√-3 1-√-3 1+√-3 1 0 0 0 0 0 0 0 complex faithful

Permutation representations of C32.S4
On 18 points - transitive group 18T98
Generators in S18
```(2 8 5)(3 6 9)(10 16 13)(11 14 17)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 12)(2 13)(4 15)(5 16)(7 18)(8 10)
(2 13)(3 14)(5 16)(6 17)(8 10)(9 11)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 12)(2 11)(3 10)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)```

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

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

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

`G:=TransitiveGroup(18,98);`

On 18 points - transitive group 18T101
Generators in S18
```(2 8 5)(3 6 9)(10 16 13)(11 14 17)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 15)(2 16)(4 18)(5 10)(7 12)(8 13)
(2 16)(3 17)(5 10)(6 11)(8 13)(9 14)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(2 9)(3 8)(4 7)(5 6)(10 11)(12 18)(13 17)(14 16)```

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

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

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

`G:=TransitiveGroup(18,101);`

C32.S4 is a maximal quotient of   C32.CSU2(𝔽3)  C32.GL2(𝔽3)  C62.Dic3

Matrix representation of C32.S4 in GL6(ℤ)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0
,
 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1
,
 0 0 0 0 -1 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 1 0 0 -1 0 0 0 0 0 -1 1 0 0

`G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,-1,0,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,-1,-1,0,0,0,0,0,1,0,0] >;`

C32.S4 in GAP, Magma, Sage, TeX

`C_3^2.S_4`
`% in TeX`

`G:=Group("C3^2.S4");`
`// GroupNames label`

`G:=SmallGroup(216,90);`
`// by ID`

`G=gap.SmallGroup(216,90);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-2,2,542,224,122,867,3244,556,1949,989]);`
`// Polycyclic`

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

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