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C₄²×ΓL(1,𝔽₈) #66579

order	336
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₄

Aliases

C₄²×ΓL(1,𝔽₈)

Splittings

Fitting, Hall	C₄²×C₇ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₃×C₄²
Frattini	C₂²	C₂²×ΓL(1,𝔽₈)
Fitting	C₄²×C₇	C₃
socle, cosocle	C₂²×C₇	C₂²×C₃
prime radical	C₂²×ΓL(1,𝔽₈)	C₂²
LCS	G ▷ C₇
UCS	C₁ ◁ C₄²

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 15 points:

(2,3,4)(5,7,6)(12,13,14,15);(1,2,3,5,4,6,7)(8,9,10,11)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×ΓL(1,𝔽₈)⋊C₈, #6377⋊C₃, C₄⋊(ΓL(1,𝔽₈)⋊C₈), …
maximal normal subgroup of	C₄×ΓL(1,𝔽₈)⋊C₈, #6377⋊C₃, C₄⋊(ΓL(1,𝔽₈)⋊C₈), …
maximal quotient of	ΓL(1,𝔽₈)×#95, C₄×C₈×ΓL(1,𝔽₈), ΓL(1,𝔽₈)×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 48);