Finite Group Databasebeta

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C₉×C₃₂×C₅₃ #72193270

order	15264
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₄×C₈×C₁₃ (order 4992) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₄×C₈×C₁₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₅₂₆₄
abelian	C₉×C₃₂×C₅₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₁₆	C₂×C₃×C₅₃
socle, prime radical	C₂×C₃×C₅₃	C₃×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

36 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

15264 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₅₂₆₄, D₃₀₅₂₈
Fitting subgroup of	D₁₅₂₆₄, Dic₇₆₃₂
maximal subgroup of	C₉×C₅₃×C₆₄, C₂×C₉×C₃₂×C₅₃, D₁₅₂₆₄, …
maximal normal subgroup of	C₉×C₅₃×C₆₄, C₂×C₉×C₃₂×C₅₃, D₁₅₂₆₄, …
maximal quotient of	C₉×C₅₃×C₆₄, C₂×C₉×C₃₂×C₅₃

GAP

Past the SmallGroups library coverage — no SmallGroups id.