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A fake quadric is a smooth minimal surface of general type with the sameinvariants as the quadric in P^3, i.e. K^2=2c_2=8 and q=p_g=0. We study herequaternionic fake quadrics i.e. fake quadrics constructed arithmetically byusing some quaternion algebras over real number fields.  We provide examples of quaternionic fake quadrics X with a non-trivialautomorphism group and compute the invariants of the minimal desingularisationof the quotient of X by this group. In that way we obtain minimal surfaces ofgeneral type Z with q=p_g=0 and K^2=4,2 which contain the maximal number ofdisjoint nodal curves. We then prove that if a surface of general type has thesame invariant as Z and same number of nodal curves, we can constructgeometrically a surface of general type with K^2=2c_2=8.

Automorphisms and quotients of quaternionic fake quadrics