C₂×C₃²×ΓL(1,𝔽₈) #67399

order	378
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product	C₂×C₃²×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₂×C₇)⋊C₃³
unambiguous semidirect product	(C₂×C₃×C₇)⋊C₃²

	subgroup	G/H
center, hypercenter	C₂×C₃²	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃³
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃²

112 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 3 generators on 15 points:

(1,2)(10,11,12)(13,15,14);(3,4,5)(9,10,11,13,12,14,15);(6,7,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 52);