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We consider nontrivial topological phases in Floquet systems using unitaryloops and stroboscopic evolutions under a static Floquet Hamiltonian $H_F$ inthe presence of dynamical space-time symmetries $G$. While the latter has beensubject of out-of-equilibrium classifications that extend the ten-fold way andsystems with additional crystalline symmetries to periodically driven systems,we explore the anomalous topological zero modes that arise in $H_F$ from thecoexistence of a dynamical space-time symmetry $M$ and antisymmetry $A$ of $G$,and classify them using a frequency-domain formulation. Moreover, we provide aninterpretation of the resulting Floquet topological phases using a frequencylattice with a decoration represented by color degrees of freedom on thelattice vertices. These colors correspond to the coefficient $N$ of the groupextension $\tilde{G}$ of $G$ along the frequency lattice, given by $N=Z\rtimesH^1[A,M]$. The distinct topological classifications that arise at differentenergy gaps in its quasi-energy spectrum are described by the torsion productof the cohomology group $H^{2}[G,N]$ classifying the group extension.

Distinct Floquet topological classifications from color-decorated frequency lattices with space-time symmetries