  The Dicke-Ising model describes cavity quantum electrodynamics setups in
which dipoles couple not only with the photonic cavity field but also to each
other through dipole-dipole interaction. In this work we diagonalise such a
model in terms of bosonic polaritonic operators for arbitrarily large values of
the light-matter coupling and for values of the dipole-dipole interaction until
the onset of the ferromagnetic Ising phase transition. In order to accomplish
this task we exploit higher order terms of the Holstein-Primakoff
transformation, developing a general method which allows to solve the normal
phase of the Ising model in term of bosonic excitations for large values of the
dipole-dipole interaction. Our results shed light on the interplay between the
dipole-dipole and the light-matter coupling strengths, and their effect on the
virtual excitations which populate the ground-state when the interactions
become comparable with the bare transition frequency.
