Finite Group Databasebeta

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C₁₆×C₂₇×C₂₉ #72190865

order	12528
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₄²×C₇×C₉ (order 4032) · Inn(G) = C₁ · Out(G) = C₂²×C₄²×C₇×C₉
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂₅₂₈
abelian	C₁₆×C₂₇×C₂₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈×C₉	C₂×C₃×C₂₉
socle, prime radical	C₂×C₃×C₂₉	C₈×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

40 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

12528 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₂₅₂₈, D₂₅₀₅₆
maximal subgroup of	C₂₇×C₂₉×C₃₂, C₂×C₁₆×C₂₇×C₂₉, Dic₆₂₆₄, …
maximal normal subgroup of	C₂₇×C₂₉×C₃₂, C₂×C₁₆×C₂₇×C₂₉, Dic₆₂₆₄, …
maximal quotient of	C₂₇×C₂₉×C₃₂, C₂×C₁₆×C₂₇×C₂₉

GAP

Past the SmallGroups library coverage — no SmallGroups id.