  We obtain a detailed classification for a class of non-simply connected
Calabi-Yau threefolds which are of potential interest for a wide range of
problems in string phenomenology. These threefolds arise as quotients of
Schoen's Calabi-Yau threefolds, which are fiber products over P1 of two
rational elliptic surfaces. The quotient is by a freely acting finite abelian
group preserving the fibrations. Our work involves a classification of
restricted finite automorphism groups of rational elliptic surfaces.
