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## G = C32order 9 = 32

### Elementary abelian group of type [3,3]

Aliases: C32, SmallGroup(9,2)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32
 Chief series C1 — C3 — C32
 Lower central C1 — C32
 Upper central C1 — C32
 Jennings C1 — C32

Generators and relations for C32
G = < a,b | a3=b3=1, ab=ba >

Character table of C32

 class 1 3A 3B 3C 3D 3E 3F 3G 3H size 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ32 1 linear of order 3 ρ3 1 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ3 1 linear of order 3 ρ4 1 ζ32 ζ32 1 ζ3 ζ32 1 ζ3 ζ3 linear of order 3 ρ5 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 linear of order 3 ρ6 1 1 ζ32 ζ32 1 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ7 1 ζ3 ζ3 1 ζ32 ζ3 1 ζ32 ζ32 linear of order 3 ρ8 1 1 ζ3 ζ3 1 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ9 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 linear of order 3

Permutation representations of C32
Regular action on 9 points - transitive group 9T2
Generators in S9
```(1 2 3)(4 5 6)(7 8 9)
(1 9 5)(2 7 6)(3 8 4)```

`G:=sub<Sym(9)| (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4)>;`

`G:=Group( (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4) );`

`G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9)], [(1,9,5),(2,7,6),(3,8,4)]])`

`G:=TransitiveGroup(9,2);`

C32 is a maximal subgroup of   C3⋊S3  He3  3- 1+2
C32 is a maximal quotient of   He3  3- 1+2

Polynomial with Galois group C32 over ℚ
actionf(x)Disc(f)
9T2x9-15x7-4x6+54x5+12x4-38x3-9x2+6x+126·312·76·1272

Matrix representation of C32 in GL2(𝔽7) generated by

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

C32 in GAP, Magma, Sage, TeX

`C_3^2`
`% in TeX`

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

`G:=SmallGroup(9,2);`
`// by ID`

`G=gap.SmallGroup(9,2);`
`# by ID`

`G:=PCGroup([2,-3,3]:ExponentLimit:=1);`
`// Polycyclic`

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

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