We present an algorithm for the efficient generation of all pairwise non-isomorphic cycle permutation graphs, i.e. cubic graphs with a 2-factor consisting of two chordless cycles, and non-hamiltonian cycle permutation graphs, from which the permutation snarks can easily be computed. This allows us to generate all cycle permutation graphs up to order 34 and all permutation snarks up to order 46, improving upon previous computational results by Brinkmann et al. Moreover, we give several improved lower bounds for interesting permutation snarks, such as for a smallest permutation snark of order \(6\bmod 8\) or a smallest permutation snark of girth at least 6. These computational results also allow us to complete a characterisation of the orders for which non-hamiltonian cycle permutation graphs exist, answering an open question by Klee from 1972, and yield many more counterexamples to a conjecture by Zhang.