order	126
isoclinism	isoclinic class of D₃×ΓL(1,𝔽₈)
automorphism	Aut(G) = D₃×AGL(1,𝔽₇) (order 252) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = D₃²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×ΓL(1,𝔽₈)
unambiguous semidirect product	C₃⋊(C₂×ΓL(1,𝔽₈))
unambiguous semidirect product	(D₃×C₇)⋊C₃

Splittings

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁

18 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)(5,6,7)(8,10,9);(1,2,3)(4,5,6,8,7,9,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 8);