C₂³×ΓL(1,𝔽₈) #4281

order	168
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = AGL(1,𝔽₇)×GL(3,𝔽₂) (order 7056) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂×GL(3,𝔽₂)
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

Fitting, socle, Hall	C₂³×C₇ ⋊ C₃
derived, Sylow	C₇ ⋊ C₂³×C₃

	subgroup	G/H
center, hypercenter	C₂³	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³

64 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 3 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 51);