Central and medial quasigroups of small order

We enumerate central and medial quasigroups of order less than \(128\) up to isomorphism, with the exception of those quasigroups that are isotopic to \(C_4\times C_2^4\), \(C_2^6\), \(C_3^4\) or \(C_5^3\). We give an explicit formula for the number of quasigroups that are affine over a finite cyclic group.