Magnetic properties and underlying magnetic models of the synthetic A$_2$Cu$_3$O(SO$_4)_3$ fedotovite (A = K) and puninite (A = Na) minerals, as well as the mixed euchlorine-type NaKCu$_3$O(SO$_4)_3$ are reported. We show that all these compounds contain magnetic Cu$_6$ hexamer units, which at temperatures below about 100 K act as single spin-1 entities. Weak interactions between these magnetic molecules lead to long-range order below $T_N$ = 3.4 K (A = Na), 4.7 K (A = NaK), and about 3.0 K (A = K). The formation of the magnetic order is elucidated by ab initio calculations that reveal two-dimensional inter-hexamer interactions within crystallographic $bc$ planes. This model indicates the presence of a weakly distorted square lattice of $S=1$ magnetic ions and challenges the earlier description of the A$_2$Cu$_3$O(SO$_4)_3$ minerals in terms of Haldane spin chains.
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