  A genuine tripartite entanglement of a mixed spin-(1/2,1) Heisenberg tetramer
is rigorously analyzed in a presence of external magnetic field. The couple of
mixed spin-(1/2,1) dimers is arranged in a perfect rectangular square plaquette
involving two nonequivalent Heisenberg exchange couplings $J$ and $J_1$. The
degree of a genuine tripartite entanglement is evaluated according to the
genuine tripartite negativity ${\cal N}_{ABC}$ defined as a geometric mean of
all possible bipartite negativities corresponding to a decomposition into a
single spin and the remaining spin dimer ${\cal N}_{A|BC}$, ${\cal N}_{B|AC}$
and ${\cal N}_{C|AB}$ after degrees of freedom of the last fourth spin $D$ are
traced out. Due to the symmetry of a mixed spin-(1/2,1) Heisenberg tetramer two
different genuine tripartite negativities for the trimeric system
$1/2\!-\!1\!-\!1$ and $1/2\!-\!1/2\!-\!1$ were identified. It was found that
the genuine tripartite negativity for the interaction ratio $J_1/J\!<\!1$
becomes nonzero solely in the tripartite system $1/2\!-\!1\!-\!1$ at low-enough
magnetic fields. The opposite interaction limit $J_1/J\!>\!1$ gives rise to the
nonzero genuine tripartite negativity in both tripartite systems in a presence
of external magnetic field until the classical ferromagnetic state is achieved.
It was shown, that the genuine tripartite negativity of a mixed spin-(1/2,1)
Heisenberg tetramer can be detected also at nonzero temperatures. An
enhancement of the thermal genuine tripartite negativity through the
enlargement of the total spin number of a tripartite system is evidenced. The
correlation between the bipartite negativity of two spins and the genuine
tripartite negativity is discussed in detail.
