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C₄×ΓL(1,𝔽₈)×C₈₉ #21418754

order	7476
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄×ΓL(1,𝔽₈)×C₈₉
unambiguous semidirect product	(C₄×C₇)⋊(C₃×C₈₉)
unambiguous semidirect product	(C₄×C₇×C₈₉)⋊C₃

Splittings

Fitting, Hall	C₄×C₇×C₈₉ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃×C₄×C₈₉
Hall	C₇×C₈₉ ⋊ C₃×C₄
Hall	C₄×C₇ ⋊ C₃×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₈₉	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₃×C₄×C₈₉
Frattini	C₂	C₂×ΓL(1,𝔽₈)×C₈₉
Fitting	C₄×C₇×C₈₉	C₃
socle	C₂×C₇×C₈₉	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₄×C₈₉

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1780 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃×ΓL(1,𝔽₈)×C₈₉, C₃×C₄×ΓL(1,𝔽₈)×C₈₉, C₄×C₅×ΓL(1,𝔽₈)×C₈₉, …
maximal normal subgroup of	C₃×C₄×ΓL(1,𝔽₈)×C₈₉, C₄×C₅×ΓL(1,𝔽₈)×C₈₉
maximal quotient of	C₄×C₇⋊C₉×C₈₉, Dic₃×ΓL(1,𝔽₈)×C₈₉, C₃×C₄×ΓL(1,𝔽₈)×C₈₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7476, 2);