Finite Group Databasebeta

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C₇×C₈×C₂₈₉ #72194142

order	16184
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂³×C₃×C₁₆×C₁₇ (order 6528) · Inn(G) = C₁ · Out(G) = C₂³×C₃×C₁₆×C₁₇
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₆₁₈₄
abelian	C₇×C₈×C₂₈₉

Subgroups

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

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

16184 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	Dic₁₆₁₈₄, D₃₂₃₆₈
Fitting subgroup of	Dic₈₀₉₂, D₁₆₁₈₄
maximal subgroup of	C₇×C₁₆×C₂₈₉, C₂×C₇×C₈×C₂₈₉, Dic₈₀₉₂, …
maximal normal subgroup of	C₇×C₁₆×C₂₈₉, C₂×C₇×C₈×C₂₈₉, Dic₈₀₉₂, …
maximal quotient of	C₇×C₁₆×C₂₈₉, C₂×C₇×C₈×C₂₈₉

GAP

Past the SmallGroups library coverage — no SmallGroups id.