order	144
isoclinism	isoclinic class of C₂²⋊A₄
automorphism	Aut(G) = D₃×AΓL(2,𝔽₄) (order 34560) · Inn(G) = C₂²⋊A₄ · Out(G) = D₃×S₅
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

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

Splittings

Fitting, socle	C₂⁴×C₃ ⋊ C₃
derived, Sylow	C₂⁴ ⋊ C₃²

	subgroup	G/H
center, hypercenter	C₃	C₂²⋊A₄
commutator, cosocle, hypocenter	C₂⁴	C₃²
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃	C₃
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₃

82 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 194);