Existence Of Regular 3-Polytopes Of Order 2(N)

In this paper, we prove that for any positive integers n, s, t such that n >= 10, s, t >= 2 and n - 1 >= s + t, there exists a regular polytope with Schlafli type {2(s), 2(t)} and its automorphism group is of order 2(n). Furthermore, we classify regular polytopes with automorphism groups of order 2(n) and Schlafli types {4,2(n-3)}, {4,2(n-4)} and {4,2(n-5)}, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Period. Math. Hangar. 53 (2006), no. 1-2, 231-255].