order	156
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃×C₄×S₄ (order 288) · Inn(G) = A₄ · Out(G) = C₂×C₃×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₁₃)×A₄⋊S₄
homology	Abelianization (H₁) = C₃×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₁₃
unambiguous semidirect product	C₂²⋊(C₃×C₁₃)

Splittings

Fitting, socle, Hall	C₂²×C₁₃ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₁₃

	subgroup	G/H
center, hypercenter	C₁₃	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₁₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₁₃

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 52

A minimal faithful permutation representation — 2 generators on 17 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17);(1,4)(2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(156, 13);