C₂²×D₃×ΓL(1,𝔽₈) #92753

order	504
isoclinism	isoclinic class of D₃×ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×D₃×ΓL(1,𝔽₈)
unambiguous semidirect product	C₃⋊(C₂³×ΓL(1,𝔽₈))
unambiguous semidirect product	(C₂×C₃)⋊(C₂²×ΓL(1,𝔽₈))

	subgroup	G/H
center, hypercenter	C₂²	D₃×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₃×C₇	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂²

138 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 3 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 179);