order	228
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₂×C₉×S₄ (order 432) · Inn(G) = A₄ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₃×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₁₉
unambiguous semidirect product	C₂²⋊(C₃×C₁₉)

Splittings

Fitting, socle, Hall	C₂²×C₁₉ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₁₉

	subgroup	G/H
center, hypercenter	C₁₉	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₂²×C₁₉	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₁₉

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 76

A minimal faithful permutation representation — 2 generators on 23 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23);(1,4)(2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(228, 10);