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C₈×C₉×C₁₃₉ #72188686

order	10008
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂⁴×C₃²×C₂₃ (order 3312) · Inn(G) = C₁ · Out(G) = C₂⁴×C₃²×C₂₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₀₀₀₈
abelian	C₈×C₉×C₁₃₉
multiplicative	(ℤ/10009)ˣ, (ℤ/20018)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄	C₂×C₃×C₁₃₉
socle, prime radical	C₂×C₃×C₁₃₉	C₃×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

10008 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₀₀₀₈, D₂₀₀₁₆
maximal subgroup of	C₉×C₁₆×C₁₃₉, C₂×C₈×C₉×C₁₃₉, C₈×C₂₇×C₁₃₉, …
maximal normal subgroup of	C₉×C₁₆×C₁₃₉, C₂×C₈×C₉×C₁₃₉, C₈×C₂₇×C₁₃₉, …
maximal quotient of	C₉×C₁₆×C₁₃₉, C₂×C₈×C₉×C₁₃₉, C₈×C₂₇×C₁₃₉, …
automorphism group of	C₁₀₀₀₉
outer automorphism group of	C₁₀₀₀₉

GAP

Past the SmallGroups library coverage — no SmallGroups id.