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G = C₈  order 8 = 2³

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C₈, also denoted Z₈, SmallGroup(8,1)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Generators and relations for C₈
 G = < a | a⁸=1 >

Character table of C₈

(1 2 3 4 5 6 7 8)

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

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

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

G:=TransitiveGroup(8,1);

C₈ is a maximal subgroup of
 C₁₆  D₈  SD₁₆  Q₁₆  C₃²⋊₂C₈  F₉  C₁₇⋊C₈  C₅²⋊C₈  C₄₁⋊C₈  C₇²⋊₂C₈  C₇²⋊C₈
 C_2p.C₄: M₄(2)  C₃⋊C₈  C₅⋊₂C₈  C₅⋊C₈  C₇⋊C₈  C₁₁⋊C₈  C₁₃⋊₂C₈  C₁₃⋊C₈ ...
C₈ is a maximal quotient of
 C₁₆  C₃²⋊₂C₈  F₉  C₅²⋊C₈  C₇²⋊₂C₈  C₇²⋊C₈  A₅⋊C₈
 C_p⋊C₈: C₃⋊C₈  C₅⋊₂C₈  C₅⋊C₈  C₇⋊C₈  C₁₁⋊C₈  C₁₃⋊₂C₈  C₁₃⋊C₈  C₁₇⋊₃C₈ ...

Matrix representation of C₈ ►in GL₁(𝔽₁₇) generated by

G:=sub<GL(1,GF(17))| [2] >;

C₈ in GAP, Magma, Sage, TeX

C_8

% in TeX

G:=Group("C8");

// GroupNames label

G:=SmallGroup(8,1);

// by ID

G=gap.SmallGroup(8,1);

# by ID

G:=PCGroup([3,-2,-2,-2,6,16]);

// Polycyclic

G:=Group<a|a^8=1>;

// generators/relations

Export

Subgroup lattice of C₈ in TeX
Character table of C₈ in TeX