Finite Group Databasebeta

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C₃×C₅×C₁₃×C₆₄ #72190818

order	12480
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₄²×C₁₆ (order 3072) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₄²×C₁₆
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂₄₈₀
abelian	C₃×C₅×C₁₃×C₆₄

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂×C₃×C₅×C₁₃
socle, prime radical	C₂×C₃×C₅×C₁₃	C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

56 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

12480 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₂₄₈₀, D₂₄₉₆₀
maximal subgroup of	C₃×C₅×C₁₃×C₁₂₈, C₂×C₃×C₅×C₁₃×C₆₄, Dic₆₂₄₀, …
maximal normal subgroup of	C₃×C₅×C₁₃×C₁₂₈, C₂×C₃×C₅×C₁₃×C₆₄, Dic₆₂₄₀, …
maximal quotient of	C₃×C₅×C₁₃×C₁₂₈, C₂×C₃×C₅×C₁₃×C₆₄

GAP

Past the SmallGroups library coverage — no SmallGroups id.