C₂×A₄×ΓL(1,𝔽₈) #92737

order	504
isoclinism	isoclinic class of A₄×ΓL(1,𝔽₈)
automorphism	Aut(G) = S₄×AGL(1,𝔽₇) (order 1008) · Inn(G) = A₄×ΓL(1,𝔽₈) · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₂×A₄×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₂³×ΓL(1,𝔽₈))⋊C₃
unambiguous semidirect product	(C₂×C₇×A₄)⋊C₃
unambiguous semidirect product	(C₂×C₂²⋊ΓL(1,𝔽₈))⋊C₃

	subgroup	G/H
center, hypercenter	C₂	A₄×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₂³×C₇	C₃²
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂

64 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 163);