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G = C₉  order 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

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

Character table of C₉

(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);

C₉ is a maximal subgroup of
 D₉  C₂₇  3_-¹⁺²  C₃.A₄  C₅²⋊C₉
 C_p⋊C₉, p=1 mod 3: C₇⋊C₉  C₁₃⋊C₉  C₁₉⋊₂C₉  C₁₉⋊C₉  C₃₁⋊C₉  C₃₇⋊₂C₉  C₃₇⋊C₉  C₄₃⋊C₉ ...
C₉ is a maximal quotient of
 C₂₇  C₃.A₄  C₅²⋊C₉
 C_p⋊C₉, p=1 mod 3: C₇⋊C₉  C₁₃⋊C₉  C₁₉⋊₂C₉  C₁₉⋊C₉  C₃₁⋊C₉  C₃₇⋊₂C₉  C₃₇⋊C₉  C₄₃⋊C₉ ...

Matrix representation of C₉ ►in GL₁(𝔽₁₉) generated by

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

Export

Subgroup lattice of C₉ in TeX
Character table of C₉ in TeX