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ΓL(1,𝔽₈)×#4747 #72254353

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

ΓL(1,𝔽₈)×#4747

Splittings

Fitting	C₃×C₇×#100 ⋊ C₂×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ #453
Sylow	C₇ ⋊ C₃×#4747

Subgroups

	subgroup	G/H
center	C₂	ΓL(1,𝔽₈)×#882
commutator, cosocle	C₂×C₃×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×D₃×ΓL(1,𝔽₈)
Fitting	C₃×C₇×#100	C₂×C₃
socle	C₂×C₃×C₇	C₃×#120
hypercenter	#100	D₃×ΓL(1,𝔽₈)
hypocenter	C₃×C₇	C₃×#453
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

718 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×ΓL(1,𝔽₈)×#453, AGL(1,𝔽₇)×#4747, #72254465
maximal normal subgroup of	D₃×ΓL(1,𝔽₈)×#453, AGL(1,𝔽₇)×#4747, #72254465

GAP

Past the SmallGroups library coverage — no SmallGroups id.