Finite Group Databasebeta

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C₂₇×C₃₁₁ #22554231

order	8397
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₅×C₉×C₃₁ (order 5580) · Inn(G) = C₁ · Out(G) = C₂²×C₅×C₉×C₃₁
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₈₃₉₇
abelian	C₂₇×C₃₁₁

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini	C₉	C₃×C₃₁₁
socle	C₃×C₃₁₁	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

8 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

8397 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₈₃₉₇, C₂×D₈₃₉₇
maximal subgroup of	C₈₁×C₃₁₁, C₃×C₂₇×C₃₁₁, C₂₇⋊C₃×C₃₁₁, …
maximal normal subgroup of	C₈₁×C₃₁₁, C₃×C₂₇×C₃₁₁, C₂₇⋊C₃×C₃₁₁, …
maximal quotient of	C₈₁×C₃₁₁, C₃×C₂₇×C₃₁₁, C₂×C₂₇×C₃₁₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8397, 1);