  Higher-order and fractional discrete time crystals (DTCs) are exotic phases
of matter where the discrete time translation symmetry is broken into
higher-order and non-integer category. Generation of these unique DTCs has been
widely studied theoretically in different systems. However, no current
experimental methods can probe these higher-order and fractional DTCs in any
quantum many-body systems. We demonstrate an experimental approach to observe
higher-order and fractional DTCs in Floquet-driven Rydberg atomic gases. We
have discovered multiple $n$-DTCs with integer values of $n$ = 2, 3, and 4, and
others ranging up to 14, along with fractional $n$-DTCs with $n$ values beyond
the integers. The system response can transition between adjacent integer DTCs,
during which the fractional DTCs are investigated. Study of higher-order and
fractional DTCs expands fundamental knowledge of non-equilibrium dynamics and
is promising for discovery of more complex temporal symmetries beyond the
single discrete time translation symmetry.
