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G = C9order 9 = 32

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C9, also denoted Z9, SmallGroup(9,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C9
C1C3 — C9
C1 — C9
C1 — C9
C1C3C3 — C9

Generators and relations for C9
 G = < a | a9=1 >


Character table of C9

 class 13A3B9A9B9C9D9E9F
 size 111111111
ρ1111111111    trivial
ρ21ζ32ζ3ζ97ζ95ζ9ζ94ζ98ζ92    linear of order 9 faithful
ρ31ζ3ζ32ζ95ζ9ζ92ζ98ζ97ζ94    linear of order 9 faithful
ρ4111ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ51ζ32ζ3ζ9ζ92ζ94ζ97ζ95ζ98    linear of order 9 faithful
ρ61ζ3ζ32ζ98ζ97ζ95ζ92ζ94ζ9    linear of order 9 faithful
ρ7111ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ81ζ32ζ3ζ94ζ98ζ97ζ9ζ92ζ95    linear of order 9 faithful
ρ91ζ3ζ32ζ92ζ94ζ98ζ95ζ9ζ97    linear of order 9 faithful

Permutation representations of C9
Regular action on 9 points - transitive group 9T1
Generators in S9
(1 2 3 4 5 6 7 8 9)

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

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

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

G:=TransitiveGroup(9,1);

Polynomial with Galois group C9 over ℚ
actionf(x)Disc(f)
9T1x9-x8-8x7+7x6+21x5-15x4-20x3+10x2+5x-1198

Matrix representation of C9 in GL1(𝔽19) generated by

4
G:=sub<GL(1,GF(19))| [4] >;

C9 in GAP, Magma, Sage, TeX

C_9
% in TeX

G:=Group("C9");
// GroupNames label

G:=SmallGroup(9,1);
// by ID

G=gap.SmallGroup(9,1);
# by ID

G:=PCGroup([2,-3,-3,6]:ExponentLimit:=1);
// Polycyclic

G:=Group<a|a^9=1>;
// generators/relations

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