C₂×C₃×ΓL(1,𝔽₈) #1261

order	126
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) = C₂×#72224676
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₂×C₃×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₂×C₇)⋊C₃²
unambiguous semidirect product	(C₂×C₃×C₇)⋊C₃

	subgroup	G/H
center, hypercenter	C₂×C₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 2 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 10);