C₂²×ΓL(1,𝔽₈) #750

order	84
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) = S₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

Fitting, socle, Hall	C₂²×C₇ ⋊ C₃
derived, Sylow	C₇ ⋊ C₂²×C₃

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

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(84, 9);