  In this work we discuss the existence of time-translation symmetry breaking
in a kicked infinite-range-interacting clean spin system described by the
Lipkin-Meshkov-Glick model. This Floquet time crystal is robust under
perturbations of the kicking protocol, its existence being intimately linked to
the underlying $\mathbb{Z}_2$ symmetry breaking of the time-independent model.
We show that the model being infinite-range and having an extensive amount of
symmetry breaking eigenstates is essential for having the time-crystal
behaviour. In particular we discuss the properties of the Floquet spectrum, and
show the existence of doublets of Floquet states which are respectively even
and odd superposition of symmetry broken states and have quasi-energies
differing of half the driving frequencies, a key essence of Floquet time
crystals. Remarkably, the stability of the time-crystal phase can be directly
analysed in the limit of infinite size, discussing the properties of the
corresponding classical phase space. Through a detailed analysis of the
robustness of the time crystal to various perturbations we are able to map the
corresponding phase diagram. We finally discuss the possibility of an
experimental implementation by means of trapped ions.
