  We investigate an unconventional symmetry in time-periodically driven
systems, the Floquet dynamical symmetry (FDS). Unlike the usual symmetries, the
FDS gives symmetry sectors that are equidistant in the Floquet spectrum and
protects quantum coherence between them from dissipation and dephasing, leading
to two kinds of time crystals: the discrete time crystal and discrete time
quasicrystal that have different periodicity in time. We show that these time
crystals appear in the Bose- and Fermi-Hubbard models under ac fields and their
periodicity can be tuned only by adjusting the strength of the field. These
time crystals arise only from the FDS and thus appear in both dissipative and
isolated systems and in the presence of disorder as long as the FDS is
respected. We discuss their experimental realizations in cold atom experiments
and generalization to the SU($N$)-symmetric Hubbard models.
