Define the following incidence structure of rank three. The points are the elements of $\mathbb{Z}_7=$ {$0,\ldots,6$}. The lines of type 1 are the triples $(x,x+1,x+3)$ modulo $7$. The lines of type 2 are the triples $(x,x+1,x+5)$ modulo 7. Define the incidence relation as follows. A point is incident to a line of type 1 (resp.2) if it is contained in the line. A line of type 1 is incident to a line of type 2 if they have two points in common.

It is not difficult to check that this incidence structure is a geometry (in the sense of Buekenhout). Somehow it looks like two superposed Fano planes.

Here is my question: what is the full automorphism group of this geometry, and what is the type preserving automorphism group of this geometry?

Thanks.