Since for each face there is another parallel to it, $n$ must be even. For $n = 4k$, attach two congruent regular pyramids with $2k$ lateral faces by its bases. For $n = 4k + 2$, attach two congruent regular pyramids with $2k + 1$ lateral faces and twist one of them. Possible coordinates for the vertices of this solid are $(\cos \frac{\pi j}{2k+1}, \sin \frac{\pi j}{2k+1}, (-1)^j(1 - \cos \frac{\pi}{2k+1}))$, $j = 1, 2, \dots, 4k + 1$ and $(0, 0, \pm (1 + \cos \frac{\pi}{2k+1}))$.