order	147
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = C₂×C₃×AGL(1,𝔽₇) (order 252) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

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

Splittings

Fitting, socle, Sylow	C₇² ⋊ C₃
derived	C₇ ⋊ C₃×C₇

	subgroup	G/H
center, hypercenter	C₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₇
Frattini	C₁	G
Fitting, socle	C₇²	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₇

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 21

A minimal faithful permutation representation — 2 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(147, 3);