Finite Group Databasebeta

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C₄×C₂₇×C₁₀₉ #72190204

order	11772
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₄×C₉×C₂₇ (order 3888) · Inn(G) = C₁ · Out(G) = C₂²×C₄×C₉×C₂₇
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₁₇₇₂
abelian	C₄×C₂₇×C₁₀₉
multiplicative	(ℤ/11881)ˣ, (ℤ/23762)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₉	C₂×C₃×C₁₀₉
socle, prime radical	C₂×C₃×C₁₀₉	C₂×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

11772 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₁₇₇₂, D₂₃₅₄₄
maximal subgroup of	C₈×C₂₇×C₁₀₉, C₂×C₄×C₂₇×C₁₀₉, D₁₁₇₇₂, …
maximal normal subgroup of	C₈×C₂₇×C₁₀₉, C₂×C₄×C₂₇×C₁₀₉, D₁₁₇₇₂, …
maximal quotient of	C₈×C₂₇×C₁₀₉, C₂×C₄×C₂₇×C₁₀₉
automorphism group of	C₁₁₈₈₁
outer automorphism group of	C₁₁₈₈₁

GAP

Past the SmallGroups library coverage — no SmallGroups id.