We analyze semidirect extensions of middle nuclei of commutative automorphic loops. We find a less complicated conditions for the semidirect construction when the middle nucleus is an odd order abelian group. We then use the description to study extensions of orders and .
In this paper we consider finite loops of specific order and we show that certain abelian groups are not isomorphic to inner mapping groups of these loops. By using our results we are able to construct a finite solvable group of order 120 which is not isomorphic to the multiplication group of a finite loop.
In this short paper, we survey the results on commutative automorphic loops and give a new construction method. Using this method, we present new classes of commutative automorphic loops of exponent with trivial center.