commutative loops; construction of loops; matrices over finite fields; quadratic extensions
Using a construction of commutative loops with metacyclic inner mapping group and trivial center described by A. Drápal, we enumerate presumably all such loops of order $pq$, for $p$ and $q$ primes.
 Drápal A.: A class of commutative loops with metacyclic inner mapping groups
. Comment. Math. Univ. Carolin. 49,3 (2008), 357–382. MR 2490433
 Jedlička P., Simon D.: Commutative automorphic loops of order $pq$. preprint.