order	144
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = C₂³×D₃² (order 288) · Inn(G) = D₃² · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×C₃⋊C₈
unambiguous semidirect product	(C₃×D₃)⋊C₈

Splittings

C₃² ⋊ C₂×C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃²
commutator, hypocenter	C₃²	C₂×C₈
Fitting, cosocle	C₃²×C₄	C₂²
socle	C₂×C₃²	C₂×C₄
prime radical	C₂×C₃×D₃	C₄
LCS	G ▷ C₃²
UCS	C₁ ◁ C₄

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 52);