the automorphism group of pg(3,2) maps lines to lines. the number of automorphisms is given by finding the number of ways of selecting 4 points that are not coplanar; this works out to 15⋅14⋅12⋅8 = 20160 = 8!/2. it turns out that the automorphism group of pg(3,2) is isomorphic to the alternating group on 8 elements a.