Finite Group Databasebeta

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C₇×C₈×C₉×C₁₇ #72187422

order	8568
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂⁴×C₃²×C₁₆ (order 2304) · Inn(G) = C₁ · Out(G) = C₂⁴×C₃²×C₁₆
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₈₅₆₈
abelian	C₇×C₈×C₉×C₁₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄	C₂×C₃×C₇×C₁₇
socle, prime radical	C₂×C₃×C₇×C₁₇	C₃×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

48 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

8568 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₈₅₆₈, D₁₇₁₃₆
maximal subgroup of	C₇×C₉×C₁₆×C₁₇, C₂×C₇×C₈×C₉×C₁₇, C₇×C₈×C₁₇×C₂₇, …
maximal normal subgroup of	C₇×C₉×C₁₆×C₁₇, C₂×C₇×C₈×C₉×C₁₇, C₇×C₈×C₁₇×C₂₇, …
maximal quotient of	C₇×C₉×C₁₆×C₁₇, C₂×C₇×C₈×C₉×C₁₇, C₇×C₈×C₁₇×C₂₇, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.