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
H
whose limit set is a generalized Apollonian gasket
Λ
H
\Lambda _H
. We design a surgery that relates
H
H
to a rational map
g
g
whose Julia set
J
g
\mathcal {J}_g
is (non-quasiconformally) homeomorphic to
Λ
H
\Lambda _H
. We show for a large class of triangulations, however, the groups of quasisymmetries of
Λ
H
\Lambda _H
and
J
g
\mathcal {J}_g
are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of
H
H
, this group is equal to the group of Möbius symmetries of
Λ
H
\Lambda _H
, which is the semi-direct product of
H
H
itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when
Λ
H
\Lambda _H
is the classical Apollonian gasket), we give a quasiregular model for the above actions which is quasiconformally equivalent to
g
g
and produces
H
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.