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C₃×ΓL(1,𝔽₈)×C₈₉ #19150474

order	5607
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₈₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×ΓL(1,𝔽₈)×C₈₉
unambiguous semidirect product	C₇⋊(C₃²×C₈₉)
unambiguous semidirect product	(C₃×C₇)⋊(C₃×C₈₉)
unambiguous semidirect product	(C₇×C₈₉)⋊C₃²
unambiguous semidirect product	(C₃×C₇×C₈₉)⋊C₃

Splittings

Fitting, socle	C₃×C₇×C₈₉ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃²×C₈₉
Hall	C₇×C₈₉ ⋊ C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₈₉	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₃²×C₈₉
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₈₉	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₈₉

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1335 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×AGL(1,𝔽₇)×C₈₉, C₃×ΓL(1,𝔽₈)⋊D₈₉, C₃×ΓL(1,𝔽₈)×D₈₉, …
maximal normal subgroup of	C₃×AGL(1,𝔽₇)×C₈₉, C₃×ΓL(1,𝔽₈)⋊D₈₉, C₃×ΓL(1,𝔽₈)×D₈₉, …
maximal quotient of	C₂×C₃×ΓL(1,𝔽₈)×C₈₉, A₄×ΓL(1,𝔽₈)×C₈₉, C₃×C₂²⋊ΓL(1,𝔽₈)×C₈₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(5607, 4);