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G = C32order 9 = 32

Elementary abelian group of type [3,3]

direct product, p-group, elementary abelian, monomial

Aliases: C32, SmallGroup(9,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C32
C1C3 — C32
C1 — C32
C1 — C32
C1 — C32

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


Character table of C32

 class 13A3B3C3D3E3F3G3H
 size 111111111
ρ1111111111    trivial
ρ21ζ321ζ3ζ3ζ3ζ32ζ321    linear of order 3
ρ31ζ31ζ32ζ32ζ32ζ3ζ31    linear of order 3
ρ41ζ32ζ321ζ3ζ321ζ3ζ3    linear of order 3
ρ51ζ3ζ32ζ3ζ321ζ321ζ3    linear of order 3
ρ611ζ32ζ321ζ3ζ3ζ32ζ3    linear of order 3
ρ71ζ3ζ31ζ32ζ31ζ32ζ32    linear of order 3
ρ811ζ3ζ31ζ32ζ32ζ3ζ32    linear of order 3
ρ91ζ32ζ3ζ32ζ31ζ31ζ32    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);

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

20
01
,
10
04
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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