  According to the Circle Packing Theorem, any triangulation of the Riemann
sphere can be realized as a nerve of a circle packing. Reflections in the dual
circles generate a Kleinian group $H$ whose limit set is an Apollonian-like
gasket $\Lambda_H$. We design a surgery that relates $H$ to a rational map $g$
whose Julia set $\mathcal{J}_g$ is (non-quasiconformally) homeomorphic to
$\Lambda_H$. We show for a large class of triangulations, however, the groups
of quasisymmetries of $\Lambda_H$ and $\mathcal{J}_g$ are isomorphic and
coincide with the corresponding groups of self-homeomorphisms. Moreover, in the
case of $H$, this group is equal to the group of M\"obius symmetries of
$\Lambda_H$, which is the semi-direct product of $H$ itself and the group of
M\"obius symmetries of the underlying circle packing. In the case of the
tetrahedral triangulation (when $\Lambda_ H$ is the classical Apollonian
gasket), we give a piecewise affine model for the above actions which is
quasiconformally equivalent to $g$ and produces $H$ by a David surgery. We also
construct a mating between the group and the map coexisting in the same
dynamical plane and show that it can be generated by Schwarz reflections in the
deltoid and the inscribed circle.
