order	144
isoclinism	isoclinic class of C₄²⋊C₃
automorphism	Aut(G) = D₃×#4626⋊C₂ (order 2304) · Inn(G) = C₄²⋊C₃ · Out(G) = D₃×D₄
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₃×C₄

Aliases

direct product	C₃×C₄²⋊C₃
unambiguous semidirect product	C₄²⋊C₃²
unambiguous semidirect product	(C₃×C₄²)⋊C₃

Splittings

Fitting	C₃×C₄² ⋊ C₃
derived, Sylow	C₄² ⋊ C₃²

	subgroup	G/H
center, hypercenter	C₃	C₄²⋊C₃
commutator, cosocle, hypocenter	C₄²	C₃²
Frattini	C₂²	C₃×A₄
Fitting	C₃×C₄²	C₃
socle	C₂²×C₃	A₄
LCS	G ▷ C₄²
UCS	C₁ ◁ C₃

26 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 36

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 68);