C₇×A₄ #751

order	84
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₂×C₃×S₄ (order 144) · Inn(G) = A₄ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×A₄⋊S₄
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×A₄
unambiguous semidirect product	C₂²⋊(C₃×C₇)

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

	subgroup	G/H
center, hypercenter	C₇	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₇

10 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(84, 10);