order	63
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = D₃×AGL(1,𝔽₇) (order 252) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂×D₃
holomorph	G ⋊ Aut(G) = #72224676
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×ΓL(1,𝔽₈)
unambiguous semidirect product	C₇⋊C₃²
unambiguous semidirect product	(C₃×C₇)⋊C₃

Splittings

Fitting, socle	C₃×C₇ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃²

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 21

A minimal faithful permutation representation — 2 generators on 10 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(63, 3);