In 1966 L. Carlitz conjectured that for every even positive integer \(n\) there exists a constant \(c_ n\) such that for any odd \(q>c_ n\) there is no permutation polynomial of degree \(n\) over the finite field \(F_ q\) of order \(q\). This conjecture was known to hold for \(n\) a power of 2 and for all even \(n\leq 16\). In this paper the conjecture is shown for all even \(n<1000\) and for every \(n\) that is twice a prime. The proof exploits the connections between permutation polynomials, exceptional polynomials, and primitive permutation groups in a skillful manner. The limitation \(n<1000\) stems from the list of all primitive groups of degree \(<1000\) compiled by \textit{J. D. Dixon} and \textit{B. Mortimer} [Math. Proc. Camb. Philos. Soc. 103, 213-238 (1988; Zbl 0646.20003)]. We note that the case where \(n\) is twice a prime was settled independently by \textit{D. Wan} [Proc. Am. Math. Soc. 110, 303-309 (1990; Zbl 0711.11050)] who used a different method. The Carlitz conjecture was recently proved in full generality by \textit{M. Fried}, \textit{R. Guralnick} and \textit{J. Saxl}.