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C₇×A₄×C₈₉ #21418784

order	7476
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₇×C₈₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×A₄×C₈₉
unambiguous semidirect product	C₂²⋊(C₃×C₇×C₈₉)
unambiguous semidirect product	(C₂²×C₈₉)⋊(C₃×C₇)

Splittings

Fitting, socle, Hall	C₂²×C₇×C₈₉ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₇×C₈₉
Hall	C₂²×C₈₉ ⋊ C₃×C₇
Hall	C₂²×C₇ ⋊ C₃×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₈₉	A₄
commutator, hypocenter	C₂²	C₃×C₇×C₈₉
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₈₉	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₇×C₈₉

20 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

2492 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	A₄×ΓL(1,𝔽₈)×C₈₉, C₃×C₇×A₄×C₈₉, C₅×C₇×A₄×C₈₉, …
maximal normal subgroup of	A₄×ΓL(1,𝔽₈)×C₈₉, C₃×C₇×A₄×C₈₉, C₅×C₇×A₄×C₈₉
maximal quotient of	C₇×C₂²⋊C₉×C₈₉, C₃×C₇×A₄×C₈₉, C₅×C₇×A₄×C₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7476, 32);