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## G = C32.A4order 108 = 22·33

### The non-split extension by C32 of A4 acting via A4/C22=C3

Aliases: C32.A4, C62.2C3, C2223- 1+2, C3.A42C3, C3.4(C3×A4), (C2×C6).4C32, SmallGroup(108,21)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C32.A4
G = < a,b,c,d,e | a3=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Character table of C32.A4

 class 1 2 3A 3B 3C 3D 6A 6B 6C 6D 6E 6F 6G 6H 9A 9B 9C 9D 9E 9F size 1 3 1 1 3 3 3 3 3 3 3 3 3 3 12 12 12 12 12 12 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 ζ32 ζ3 1 ζ3 ζ3 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 linear of order 3 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ5 1 1 1 1 ζ3 ζ32 1 ζ32 ζ32 ζ32 1 ζ3 ζ3 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 linear of order 3 ρ6 1 1 1 1 ζ32 ζ3 1 ζ3 ζ3 ζ3 1 ζ32 ζ32 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 linear of order 3 ρ7 1 1 1 1 ζ3 ζ32 1 ζ32 ζ32 ζ32 1 ζ3 ζ3 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 linear of order 3 ρ8 1 1 1 1 ζ3 ζ32 1 ζ32 ζ32 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 linear of order 3 ρ9 1 1 1 1 ζ32 ζ3 1 ζ3 ζ3 ζ3 1 ζ32 ζ32 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 linear of order 3 ρ10 3 -1 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ11 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -3-3√-3/2 0 0 0 -3+3√-3/2 0 0 0 0 0 0 0 0 0 complex lifted from 3- 1+2 ρ12 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 ζ6 2 -1+√-3 -1-√-3 ζ65 2 -1-√-3 -1+√-3 0 0 0 0 0 0 complex faithful ρ13 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 ζ6 -1+√-3 -1-√-3 2 ζ65 -1-√-3 -1+√-3 2 0 0 0 0 0 0 complex faithful ρ14 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 -1 ζ65 ζ65 ζ65 -1 ζ6 ζ6 ζ6 0 0 0 0 0 0 complex lifted from C3×A4 ρ15 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 ζ6 -1-√-3 2 -1+√-3 ζ65 -1+√-3 2 -1-√-3 0 0 0 0 0 0 complex faithful ρ16 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 ζ65 2 -1-√-3 -1+√-3 ζ6 2 -1+√-3 -1-√-3 0 0 0 0 0 0 complex faithful ρ17 3 3 -3+3√-3/2 -3-3√-3/2 0 0 -3+3√-3/2 0 0 0 -3-3√-3/2 0 0 0 0 0 0 0 0 0 complex lifted from 3- 1+2 ρ18 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 ζ65 -1+√-3 2 -1-√-3 ζ6 -1-√-3 2 -1+√-3 0 0 0 0 0 0 complex faithful ρ19 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 ζ65 -1-√-3 -1+√-3 2 ζ6 -1+√-3 -1-√-3 2 0 0 0 0 0 0 complex faithful ρ20 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 -1 ζ6 ζ6 ζ6 -1 ζ65 ζ65 ζ65 0 0 0 0 0 0 complex lifted from C3×A4

Permutation representations of C32.A4
On 18 points - transitive group 18T47
Generators in S18
```(2 8 5)(3 6 9)(11 17 14)(12 15 18)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 16)(3 18)(4 10)(6 12)(7 13)(9 15)
(1 16)(2 17)(4 10)(5 11)(7 13)(8 14)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)```

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

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

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

`G:=TransitiveGroup(18,47);`

C32.A4 is a maximal subgroup of
C32.S4  C62.13C32  C62.15C32  He3.A4  He32A4  C62.C32  3- 1+2⋊A4  C62.6C32  C332A4  C62.25C32  He3.2A4  A4×3- 1+2  C62.9C32  C122.C3  C24⋊3- 1+2  C62.A4
C32.A4 is a maximal quotient of
Q8⋊3- 1+2  C62.11C32  C62⋊C9  C122.C3  C24⋊3- 1+2  C62.A4

Matrix representation of C32.A4 in GL3(𝔽7) generated by

 4 0 0 0 2 0 0 0 1
,
 2 0 0 0 2 0 0 0 2
,
 6 0 0 0 1 0 0 0 6
,
 6 0 0 0 6 0 0 0 1
,
 0 0 4 4 0 0 0 1 0
`G:=sub<GL(3,GF(7))| [4,0,0,0,2,0,0,0,1],[2,0,0,0,2,0,0,0,2],[6,0,0,0,1,0,0,0,6],[6,0,0,0,6,0,0,0,1],[0,4,0,0,0,1,4,0,0] >;`

C32.A4 in GAP, Magma, Sage, TeX

`C_3^2.A_4`
`% in TeX`

`G:=Group("C3^2.A4");`
`// GroupNames label`

`G:=SmallGroup(108,21);`
`// by ID`

`G=gap.SmallGroup(108,21);`
`# by ID`

`G:=PCGroup([5,-3,-3,-3,-2,2,45,121,1083,2029]);`
`// Polycyclic`

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

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