  Prethermalization, where quasi-steady states are realized in the intermediate
long time regime (prethermal regime), in periodically driven (Floquet) systems
is an important phenomenon since it provides a platform of nontrivial Floquet
many-body physics. In this Letter, we consider Floquet systems with dual energy
scales: the Hamiltonian consists of two different terms whose amplitude is
either comparable or much smaller than the frequency. As a result, when the
larger-amplitude drive induces a $\mathbb{Z}_N$ symmetry operation, we obtain
the effective static Hamiltonian respecting a new emergent $\mathbb{Z}_N$
symmetry in high frequency expansions, which describes the dynamics of such
Floquet systems in the prethermal regime. As an application of our formulation,
we consider prethermal discrete time crystals, in which our formalism gives a
general way to analyze them in the prethermal regime in terms of the static
effective Hamiltonian. We also provide an application to Floquet engineering,
with which we can perform simultaneous control of phases and symmetries of the
systems. This enables us to control symmetry protected topological phases even
when the original system does not respect the symmetry.
