order	168
isoclinism	isoclinic class of AGL(1,𝔽₈)
automorphism	Aut(G) = C₂×AΓL(1,𝔽₈) (order 336) · Inn(G) = AGL(1,𝔽₈) · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = D₃×#72214179
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×AGL(1,𝔽₈)
unambiguous semidirect product	(C₂³×C₃)⋊C₇

Splittings

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

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

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 44);