Searching for small simple automorphic loops

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no non-Associative simple commutative automorphic loop of order less than 212, and no non-Associative simple automorphic loop of order less than 2500. We obtain numerous examples of non-Associative simple right automorphic loops. We also prove that every automorphic loop has the antiautomorphic inverse property, and that a right automorphic loop is automorphic if and only if its conjugations are automorphisms.