On the automorphism group of a binary q-analog of the Fano plane

The smallest set of admissible parameters of a q-analog of a Steiner system is S2[2,3,7]. The existence of such a Steiner system-known as a binary q-analog of the Fano plane-is still open. In this article, the automorphism group of a putative binary q-analog of the Fano plane is investigated by a combination of theoretical and computational methods. As a conclusion, it is either rigid or its automorphism group is cyclic of order 2, 3 or 4. Up to conjugacy in GL(7,2), there remains a single possible group of order 2 and 4, respectively, and two possible groups of order 3. For the automorphisms of order 2, we give a more general result which is valid for any binary q-Steiner triple system.