Finite Group Databasebeta

← Prev (#72190851) Next (#72190853) →

C₃×C₈×C₅₂₁ #72190852

order	12504
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂³×C₅×C₈×C₁₃ (order 4160) · Inn(G) = C₁ · Out(G) = C₂³×C₅×C₈×C₁₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂₅₀₄
abelian	C₃×C₈×C₅₂₁

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₃×C₅₂₁
socle, prime radical	C₂×C₃×C₅₂₁	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

16 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

12504 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₂₅₀₄, D₂₅₀₀₈
maximal subgroup of	C₃×C₁₆×C₅₂₁, C₂×C₃×C₈×C₅₂₁, Dic₆₂₅₂, …
maximal normal subgroup of	C₃×C₁₆×C₅₂₁, C₂×C₃×C₈×C₅₂₁, Dic₆₂₅₂, …
maximal quotient of	C₃×C₁₆×C₅₂₁, C₂×C₃×C₈×C₅₂₁

GAP

Past the SmallGroups library coverage — no SmallGroups id.