Finite Group Databasebeta

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C₁₇×C₂₃×C₃₂ #72190855

order	12512
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₈×C₁₁×C₁₆ (order 5632) · Inn(G) = C₁ · Out(G) = C₂²×C₈×C₁₁×C₁₆
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂₅₁₂
abelian	C₁₇×C₂₃×C₃₂

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂×C₁₇×C₂₃
socle, prime radical	C₂×C₁₇×C₂₃	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

12512 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₂₅₁₂, D₂₅₀₂₄
maximal subgroup of	C₁₇×C₂₃×C₆₄, C₂×C₁₇×C₂₃×C₃₂, Dic₆₂₅₆, …
maximal normal subgroup of	C₁₇×C₂₃×C₆₄, C₂×C₁₇×C₂₃×C₃₂, Dic₆₂₅₆, …
maximal quotient of	C₁₇×C₂₃×C₆₄, C₂×C₁₇×C₂₃×C₃₂

GAP

Past the SmallGroups library coverage — no SmallGroups id.