Finite Group Databasebeta

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C₂₇×C₅₁₂ #72191982

order	13824
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₉×C₁₂₈ (order 4608) · Inn(G) = C₁ · Out(G) = C₂²×C₉×C₁₂₈
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₃₈₂₄
abelian	C₂₇×C₅₁₂

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉×C₂₅₆	C₂×C₃
socle, prime radical	C₂×C₃	C₉×C₂₅₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

40 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

13824 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₃₈₂₄, D₂₇₆₄₈
Fitting subgroup of	D₁₃₈₂₄, Dic₆₉₁₂
maximal subgroup of	C₂₇×C₁₀₂₄, C₂×C₂₇×C₅₁₂, D₁₃₈₂₄, …
maximal normal subgroup of	C₂₇×C₁₀₂₄, C₂×C₂₇×C₅₁₂, D₁₃₈₂₄, …
maximal quotient of	C₂₇×C₁₀₂₄, C₂×C₂₇×C₅₁₂

GAP

Past the SmallGroups library coverage — no SmallGroups id.