Periodically kicked Floquet systems, such as the kicked rotor, are a paradigmatic and illustrative simple model of chaos. For non-integrable quantum dynamics, there are several diagnostic measures, such as Loschmidt echo, autocorrelation function and out of time order correlator (OTOC) to study the presence of (or the transition to) chaotic behaviour. We analytically compute these measures in terms of the eigensystem of the unitary Floquet operator of the driven quantum systems. We use these expressions to determine the time variation of the measures for the quantum-kicked rotor (QKR) on the torus, for the integrable as well as the chaotic case. For a simpler integrable variant of the kicked rotor, we also give a representation theoretic derivation of its dynamics.