Finite Group Databasebeta

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C₅×C₁₃×C₁₂₈ #72187235

order	8320
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₄²×C₃₂ (order 3072) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₄²×C₃₂
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₈₃₂₀
abelian	C₅×C₁₃×C₁₂₈

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₆₄	C₂×C₅×C₁₃
socle, prime radical	C₂×C₅×C₁₃	C₆₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

32 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

8320 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₈₃₂₀, D₁₆₆₄₀
maximal subgroup of	C₅×C₁₃×C₂₅₆, C₂×C₅×C₁₃×C₁₂₈, C₃×C₅×C₁₃×C₁₂₈, …
maximal normal subgroup of	C₅×C₁₃×C₂₅₆, C₂×C₅×C₁₃×C₁₂₈, C₃×C₅×C₁₃×C₁₂₈, …
maximal quotient of	C₅×C₁₃×C₂₅₆, C₂×C₅×C₁₃×C₁₂₈, C₃×C₅×C₁₃×C₁₂₈

GAP

Past the SmallGroups library coverage — no SmallGroups id.