  A ferromagnetic coupling between localized Mn spins was predicted in a series
of \textit{ab initio} and tight binding calculations and experimentally
verified for the dilute magnetic semiconductor Ga$_{1-x}$Mn$_x$N. In the limit
of small Mn concentrations, $x \lesssim 0.01$, the paramagnetic properties of
this material were successfully described using a single ion crystal field
model approach. In order to obtain the description of magnetization in (Ga,Mn)N
in the presence of interacting magnetic centers, we extend the previous model
of a single substitutional Mn$^{3+}$ ion in GaN by considering pairs, triplets
and quartets of Mn$^{3+}$ ions coupled by a ferromagnetic superexchange
interaction. Using this approach we investigate how the magnetic properties,
particularly the magnitude of the uniaxial anisotropy field, change as the
number of magnetic Mn$^{3+}$ ions in a given cluster increases from 1 to 4. Our
simulations are then exploited in explaining experimental magnetic properties
of Ga$_{1-x}$Mn$_x$N with $x \cong 0.03$, where the presence of small magnetic
clusters gains in significance. As a result the approximate lower and upper
limits for the values of exchange couplings between Mn$^{3+}$ ions in GaN,
being in nearest neighbors $J_{\mathrm{nn}}$ and next nearest neighbors
$J_{\mathrm{nnn}}$ positions, respectively, are established.
