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D₃×ΓL(1,𝔽₈)×C₆₇ #22554763

order	8442
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₆₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×ΓL(1,𝔽₈)×C₆₇
unambiguous semidirect product	C₃⋊(C₂×ΓL(1,𝔽₈)×C₆₇)
unambiguous semidirect product	(D₃×C₇)⋊(C₃×C₆₇)

Splittings

Fitting, socle	C₃×C₇×C₆₇ ⋊ C₂×C₃
derived	C₃×C₇ ⋊ C₂×C₃×C₆₇
Hall	C₃×ΓL(1,𝔽₈)×C₆₇ ⋊ C₂
Hall	C₇×C₆₇ ⋊ C₃×D₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×C₆₇
Sylow	C₇ ⋊ C₃×D₃×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₆₇	D₃×ΓL(1,𝔽₈)
commutator, hypocenter	C₃×C₇	C₂×C₃×C₆₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₆₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₆₇

36 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1005 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₇)×C₆₇, D₃×ΓL(1,𝔽₈)⋊D₆₇, D₃×ΓL(1,𝔽₈)×D₆₇, …
maximal normal subgroup of	D₃×AGL(1,𝔽₇)×C₆₇, D₃×ΓL(1,𝔽₈)⋊D₆₇, D₃×ΓL(1,𝔽₈)×D₆₇, …
maximal quotient of	Dic₃×ΓL(1,𝔽₈)×C₆₇, C₂×D₃×ΓL(1,𝔽₈)×C₆₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8442, 50);