Finite Group Databasebeta

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C₇×D₇×C₈₉ #22576569

order	8722
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₇×C₈₉
unambiguous semidirect product	C₇⋊(C₂×C₇×C₈₉)

Splittings

Fitting, socle, Hall	C₇²×C₈₉ ⋊ C₂
derived	C₇ ⋊ C₂×C₇×C₈₉
Sylow	C₇² ⋊ C₂×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₈₉	D₇
commutator, hypocenter	C₇	C₂×C₇×C₈₉
Frattini	C₁	G
Fitting, socle	C₇²×C₈₉	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₇×C₈₉

22 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

3115 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×D₇×D₈₉, D₇²×C₈₉, D₇×D₆₂₃, …
maximal normal subgroup of	C₇×D₇×D₈₉, D₇²×C₈₉, D₇×D₆₂₃, …
maximal quotient of	C₇×Dic₇×C₈₉, C₂×C₇×D₇×C₈₉, C₃×C₇×D₇×C₈₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8722, 7);