Finite Group Databasebeta

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C₃×C₅×C₈×C₈₉ #72189233

order	10680
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂³×C₄×C₈×C₁₁ (order 2816) · Inn(G) = C₁ · Out(G) = C₂³×C₄×C₈×C₁₁
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₀₆₈₀
abelian	C₃×C₅×C₈×C₈₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₃×C₅×C₈₉
socle, prime radical	C₂×C₃×C₅×C₈₉	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

32 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

10680 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₀₆₈₀, D₂₁₃₆₀
maximal subgroup of	C₃×C₅×C₁₆×C₈₉, C₂×C₃×C₅×C₈×C₈₉, C₅×C₈×C₉×C₈₉, …
maximal normal subgroup of	C₃×C₅×C₁₆×C₈₉, C₂×C₃×C₅×C₈×C₈₉, C₅×C₈×C₉×C₈₉, …
maximal quotient of	C₃×C₅×C₁₆×C₈₉, C₂×C₃×C₅×C₈×C₈₉, C₅×C₈×C₉×C₈₉, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.