Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group (ℤ/a ⋊ ℤ/b) × SL2 (p)

Let G = (ℤ/a ⋊ ℤ/b) × SL2(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 ε(X(2dn − 1)/μ) of self homotopy equivalences of space forms X(2dn − 1)/μ associated with free and cellular G-actions μ on X(2dn − 1) are determined as well.