Finite Group Databasebeta

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C₅×C₈×C₃₁₇ #72191035

order	12680
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₄²×C₇₉ (order 5056) · Inn(G) = C₁ · Out(G) = C₂²×C₄²×C₇₉
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂₆₈₀
abelian	C₅×C₈×C₃₁₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₅×C₃₁₇
socle, prime radical	C₂×C₅×C₃₁₇	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

16 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

12680 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₂₆₈₀, D₂₅₃₆₀
Fitting subgroup of	Dic₆₃₄₀
maximal subgroup of	C₅×C₁₆×C₃₁₇, C₂×C₅×C₈×C₃₁₇, D₁₂₆₈₀, …
maximal normal subgroup of	C₅×C₁₆×C₃₁₇, C₂×C₅×C₈×C₃₁₇, D₁₂₆₈₀, …
maximal quotient of	C₅×C₁₆×C₃₁₇, C₂×C₅×C₈×C₃₁₇

GAP

Past the SmallGroups library coverage — no SmallGroups id.