  Various exotic topological phases of Floquet systems have been shown to arise
from crystalline symmetries. Yet, a general theory for Floquet topology that is
applicable to all crystalline symmetry groups is still in need. In this work,
we propose such a theory for (effectively) non-interacting Floquet crystals. We
first introduce quotient winding data to classify the dynamics of the Floquet
crystals with equivalent symmetry data, and then construct dynamical symmetry
indicators (DSIs) to sufficiently indicate the "inherently dynamical" Floquet
crystals. The DSI and quotient winding data, as well as the symmetry data, are
all computationally efficient since they only involve a small number of Bloch
momenta. We demonstrate the high efficiency by computing all elementary DSI
sets for all spinless and spinful plane groups using the mathematical theory of
monoid, and find a large number of different nontrivial classifications, which
contain both first-order and higher-order 2+1D anomalous Floquet topological
phases. Using the framework, we further find a new 3+1D anomalous Floquet
second-order topological insulator (AFSOTI) phase with anomalous chiral hinge
modes.
