  We consider the cosmological dynamics of the cubic Galileon field model
interacting with dark matter, where the interaction between the cosmic dark
sectors is proportional to Hubble parameter and dark energy density. The
background field equations are converted into an autonomous dynamical system of
first order equations using suitable dimensionless variables. The fixed points
are determined and their stability analyzed by means of the eigenvalue method.
The analysis of the stability of the fixed points along that of the avoidance
of ghosts and Laplacian instabilities implies negative coupling parameter. We
find an attractor fixed point in the de Sitter phase, and construct an
approximated tracker solution, which to a good approximation, mimics the
background cosmic evolution obtained by solving numerically the autonomous
system of equations. In addition, we perform the statefinder and Om diagnostics
and show that for some appropriate values of the coupling constant and initial
conditions, the trajectories in the statefinder parameter phase space evolves
towards a stable state corresponding to {\Lambda} CDM model. The comparison
with the observed distance modulus and Hubble parameter data sets clearly
indicates that the cosmological dynamics of the interacting cubic model is
compatible with the existence of an interaction between the dark sectors.
