order	288
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = C₂²×C₄×D₃² (order 576) · Inn(G) = D₃² · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×C₃⋊C₁₆
unambiguous semidirect product	(C₃×D₃)⋊C₁₆

Splittings

C₃² ⋊ C₂×C₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₃²
commutator, hypocenter	C₃²	C₂×C₁₆
Fitting, cosocle	C₃²×C₈	C₂²
socle	C₂×C₃²	C₂×C₈
prime radical	C₂×C₃×D₃	C₈
LCS	G ▷ C₃²
UCS	C₁ ◁ C₈

61 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 189);