Finite Group Databasebeta

← Prev (#20269248) Next (#20269250) →

D₃×ΓL(1,𝔽₈)×C₅₃ #20269249

order	6678
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×ΓL(1,𝔽₈)×C₅₃
unambiguous semidirect product	C₃⋊(C₂×ΓL(1,𝔽₈)×C₅₃)
unambiguous semidirect product	(D₃×C₇)⋊(C₃×C₅₃)
unambiguous semidirect product	(D₃×C₇×C₅₃)⋊C₃

Splittings

Fitting, socle	C₃×C₇×C₅₃ ⋊ C₂×C₃
derived	C₃×C₇ ⋊ C₂×C₃×C₅₃
Hall	C₃×ΓL(1,𝔽₈)×C₅₃ ⋊ C₂
Hall	C₇×C₅₃ ⋊ C₃×D₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×C₅₃
Sylow	C₇ ⋊ C₃×D₃×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₃×ΓL(1,𝔽₈)
commutator, hypocenter	C₃×C₇	C₂×C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₅₃	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₅₃

36 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

795 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₇)×C₅₃, D₃×ΓL(1,𝔽₈)⋊D₅₃, D₃×ΓL(1,𝔽₈)×D₅₃, …
maximal normal subgroup of	D₃×AGL(1,𝔽₇)×C₅₃, D₃×ΓL(1,𝔽₈)⋊D₅₃, D₃×ΓL(1,𝔽₈)×D₅₃, …
maximal quotient of	Dic₃×ΓL(1,𝔽₈)×C₅₃, C₂×D₃×ΓL(1,𝔽₈)×C₅₃, C₅×D₃×ΓL(1,𝔽₈)×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6678, 19);