We construct (for each odd prime p) q 4-dimensional regular abstract polytope P, having tetrahedral cells and vertex figures isomorphic to a regular map of type \(\{\) 3,p\(\}\) (recently described by McMullen). We also give a realization for P in whch the vertices are all \(p^ 3- \epsilon p\) points on an affine quadratic in a certain 4-dimensional orthogonal space over GF(p); here \(\epsilon =\pm 1\) for \(p\equiv \pm 1(mod 4)\). The author morphism group for P consists of all isometries in \(GO^{\epsilon}_ 4(p)\) having square spinor norm. As a by-product, we obtain for \(3<p\leq 31\) and \(p\equiv -1(mod 4)\), a new presentation for \(PSL_ 2(p^ 2):\) \[ a^ 3=b^ 3=c^ p=(ab)^ 2=(bc)^ 2=(abc)^ 2=[c^ 4(bc)c^{(p+1)/2}bc]^ 2=1 \] (which probably works for \(p>31\), too).