Discrete Floquet time crystals (DFTCs) are characterized by the spontaneous breaking of the discrete time-translational invariance characteristic of Floquet-driven systems. In analogy with equilibrium critical points, also time-crystalline phases display critical behavior of different order, i.e., oscillations whose period is a multiple $p>2$ of the Floquet driving period. Here, we introduce an experimentally accessible order parameter which is able to unambiguously detect crystalline phases regardless of the value of $p$ and, at the same time, is a useful tool for chaos diagnostic. This p aradigm allows us to investigate the phase diagram of the long-range (LR) kicked Ising model to an unprecedented depth, unveiling a rich landscape characterized by self-similar fractal boundaries. Our theoretical picture describes the emergence of DFTCs phase both as a function of the strength and period of the Floquet drive, capturing the emergent ${\mathbb{Z}}_p$ symmetry in the Floquet-Bloch waves.

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• Dynamical phase transitions
• Long-range interactions
• Time crystals
• Spin lattice models