I've got ten (projective) planes in projective 3-space:

\begin{align} &x=0\\ &z=0\\ &t=0\\ &x+y=0\\ &x-y=0\\ &z+t=0\\ &x-y-z=0\\ &x+y+z=0\\ &x-y+t=0\\ &x+y-t=0 \end{align} If I did not make a mistake somewhere, their intersections produce $25$ lines and $15$ points, each line containing $3$ points, each plane containing $6$ lines and $7$ points. Ten of the points belong to $4$ lines and $4$ planes each and five of them belong to $7$ lines and $6$ planes each. $15$ lines belong to $2$ of the planes and $10$ of them belong to $3$ of the planes.

The above indicates that this configuration is highly symmetric, is it known? How to compute its automorphism group? Where to look? My goal is to find another more symmetric realization.