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C₃×C₁₆×C₁₀₉ #72184583

order	5232
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₄²×C₂₇ (order 1728) · Inn(G) = C₁ · Out(G) = C₂²×C₄²×C₂₇
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₅₂₃₂
abelian	C₃×C₁₆×C₁₀₉
multiplicative	(ℤ/5233)ˣ, (ℤ/10466)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₃×C₁₀₉
socle, prime radical	C₂×C₃×C₁₀₉	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

20 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

5232 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₃₂×C₁₀₉, C₂×C₃×C₁₆×C₁₀₉, C₉×C₁₆×C₁₀₉, …
maximal normal subgroup of	C₃×C₃₂×C₁₀₉, C₂×C₃×C₁₆×C₁₀₉, C₉×C₁₆×C₁₀₉, …
maximal quotient of	C₃×C₃₂×C₁₀₉, C₂×C₃×C₁₆×C₁₀₉, C₉×C₁₆×C₁₀₉, …
automorphism group of	C₅₂₃₃, C₂×C₅₂₃₃
outer automorphism group of	C₅₂₃₃, C₂×C₅₂₃₃

GAP

Past the SmallGroups library coverage — no SmallGroups id.