  An ergodic system subjected to an external periodic drive will be generically
heated to infinite temperature. However, if the applied frequency is larger
than the typical energy scale of the local Hamiltonian, this heating stops
during a prethermal period that extends exponentially with the frequency.
During this prethermal period, the system may manifest an emergent symmetry
that, if spontaneously broken, will produce sub-harmonic oscillation of the
discrete time crystal (DTC). We study the role of dissipation on the survival
time of the prethermal DTC. On one hand, a bath coupling increases the
prethermal period by slowing down the accumulation of errors that eventually
destroy prethermalization. On the other hand, the spontaneous symmetry breaking
is destabilized by interaction with environment. The result of this competition
is a non-monotonic variation, i.e. the survival time of the prethermal DTC
first increases and then decreases as the environment coupling gets stronger.
