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C₃×C₇×C₁₉×ΓL(1,𝔽₈) #22552733

order	8379
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₇×C₁₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×C₇×C₁₉×ΓL(1,𝔽₈)
unambiguous semidirect product	C₇⋊(C₃²×C₇×C₁₉)
unambiguous semidirect product	(C₃×C₇)⋊(C₃×C₇×C₁₉)

Splittings

Fitting, socle	C₃×C₇²×C₁₉ ⋊ C₃
derived	C₇ ⋊ C₃²×C₇×C₁₉
Hall	C₇²×C₁₉ ⋊ C₃²
Sylow	C₇² ⋊ C₃²×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇×C₁₉	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₃²×C₇×C₁₉
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₁₉	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₇×C₁₉

56 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1995 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₇×C₁₉×AGL(1,𝔽₇), C₃×C₇×ΓL(1,𝔽₈)⋊D₁₉, C₃×C₇×ΓL(1,𝔽₈)×D₁₉, …
maximal normal subgroup of	C₃×C₇×C₁₉×AGL(1,𝔽₇), C₃×C₇×ΓL(1,𝔽₈)⋊D₁₉, C₃×C₇×ΓL(1,𝔽₈)×D₁₉, …
maximal quotient of	C₂×C₃×C₇×C₁₉×ΓL(1,𝔽₈), C₇×A₄×C₁₉×ΓL(1,𝔽₈), C₃×C₇×C₁₉×C₂²⋊ΓL(1,𝔽₈)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8379, 45);