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C₇×ΓL(1,𝔽₈)×C₁₉⋊C₃ #22552717

order	8379
automorphism	?
homology	Abelianization (H₁) = C₃²×C₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₇×ΓL(1,𝔽₈)×C₁₉⋊C₃
unambiguous semidirect product	(C₁₉×ΓL(1,𝔽₈))⋊(C₃×C₇)
unambiguous semidirect product	(C₇×C₁₉⋊C₃)⋊(C₃×C₇)
unambiguous semidirect product	#87704⋊(C₃×C₇)
unambiguous semidirect product	#87705⋊(C₃×C₇)

Splittings

Fitting, socle, Hall	C₇²×C₁₉ ⋊ C₃²
derived	C₇×C₁₉ ⋊ C₃²×C₇
Sylow	C₇² ⋊ C₃×C₁₉⋊C₃
Sylow	C₁₉ ⋊ C₃×C₇×ΓL(1,𝔽₈)

Subgroups

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

56 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

315 conjugacy classes — view the full character table.

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8379, 29);