We construct, for all $d\geq 4$, a cellulation of $\mathbb S^{d-1}$. We prove that these cellulations cannot be polytopal with maximal combinatorial symmetry. Such non-realizability phenomenon was first described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and, to the knowledge of the author, until now there have not been any known examples in higher dimensions. As a starting point for the construction, we introduce a new class of (Wythoffian) uniform polytopes, which we call duplexes. In proving our main result, we use some tools that we developed earlier while studying perfect polytopes. In particular, we prove perfectness of the duplexes; furthermore, we prove and make use of the perfectness of another new class of polytopes which we obtain by a variant of the so-called $E$-construction introduced by Eppstein, Kuperberg and Ziegler.

Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times \SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\Aut (G)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$ is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as well.