order	132
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₂×C₅×S₄ (order 240) · Inn(G) = A₄ · Out(G) = C₂²×C₅
homology	Abelianization (H₁) = C₃×C₁₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₁×A₄
unambiguous semidirect product	C₂²⋊(C₃×C₁₁)
unambiguous semidirect product	(C₂²×C₁₁)⋊C₃

Splittings

Fitting, socle, Hall	C₂²×C₁₁ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₁₁

	subgroup	G/H
center, hypercenter	C₁₁	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₁₁
Frattini	C₁	G
Fitting, socle	C₂²×C₁₁	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₁₁

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 44

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(132, 6);