  In the paper a realization of representation of a Lie algebra
$\mathfrak{sp}_4$ in the space of function on the Lie group $Sp_4$ is
considered. We find a function corresponding to a Gelfand-Tsetlin type vector
for $\mathfrak{sp}_4$ constructed by D.P. Zhelobenko. This function is
expressed though a $A$-hypergeometric function. After developing some new
technique we derive analytically formulas for the action of generators of
algebra in this base (the were not known before). These formula turn out to be
much more difficult than the formulas for the action of generators in the
Gelfand-Tsetlin type base constructed by Molev.
