A locally projective amalgam is formed by the stabilizer G(x) of a vertex x and the global stabilizer G{x,y} of an edge containing x in a group G, acting faithfully and locally finitely on a connected graph Γ of valency 2n−1 so that (i) the action is 2-arc-transitive, (ii) the sub-constituent G(x)Γ(x) is the linear group SLn(2) ≅ Ln(2) in its natural doubly transitive action, and (iii) [t,G{x,y}] ≤ O2(G(x) ∩ G{x,y}) for some t ∈ G{x,y} \ G(x). Djoković and Miller used the classical Tutte theorem to show that there are seven locally projective amalgams for n=2. Trofimov's theorem was used by the first author and Shpectorov to extend the classification to the case n≥ 3. It turned out that for n≥3, besides two infinite series of locally projective amalgams (embedded into the groups AGLn(2) and O2n+(2)), there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M22, M23, Co2, J4 and BM. For a locally projective amalgam A, the minimal degree m=m(A) of its complex representation (which is a faithful completion into GLm(C)) is calculated. The minimal representations are analysed and three open questions on exceptional locally projective amalgams are answered. It is shown that A4(1) possesses SL20(13) as a faithful completion in which the third geometric subgroup is improper; A4(2) possesses the alternating group Alt64 as a completion constrained at levels 2 and 3; A4(5) possesses Alt256 as a completion which is constrained at level 2 but not at level 3.