On the automorphism groups of smooth Fano threefolds

Let X be a smooth Fano threefold over the complex numbers of Picard rank 1 with finite automorphism group. We give numerical restrictions on the order of the automorphism group Aut (X) provided the genus g (X) 10 and X is not an ordinary smooth Gushel-Mukai threefold. More precisely, we show that the order |Aut (X) | divides a certain explicit number depending on the genus of X. We use a classification of Fano threefolds in terms of complete intersections in homogeneous varieties and the previous paper of A. Gorinov and the author regarding the topology of spaces of regular sections.