Finite Group Databasebeta

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C₁₆×C₂₅×C₂₇ #72189332

order	10800
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₄²×C₅×C₉ (order 2880) · Inn(G) = C₁ · Out(G) = C₂²×C₄²×C₅×C₉
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₀₈₀₀
abelian	C₁₆×C₂₅×C₂₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₅×C₈×C₉	C₂×C₃×C₅
socle, prime radical	C₂×C₃×C₅	C₅×C₈×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

60 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

10800 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₀₈₀₀, D₂₁₆₀₀
maximal subgroup of	C₂₅×C₂₇×C₃₂, C₂×C₁₆×C₂₅×C₂₇, C₁₆×C₂₅×C₈₁, …
maximal normal subgroup of	C₂₅×C₂₇×C₃₂, C₂×C₁₆×C₂₅×C₂₇, C₁₆×C₂₅×C₈₁, …
maximal quotient of	C₂₅×C₂₇×C₃₂, C₂×C₁₆×C₂₅×C₂₇, C₁₆×C₂₅×C₈₁, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.