Classifying non-splitting fiber preserving actions on prism manifolds

In this paper, we classify the finite groups of fiber preserving isometries G   which act on a prism manifold M(b,d)M(b,d) and do not leave a Heegaard Klein bottle invariant. We construct the following groups of isometries Z3Z3, Z6Z6, Dih(Z3)Dih(Z3) and Dih(Z6)Dih(Z6), which act on M(b,2)M(b,2) preserving the longitudinal fibering and not leaving any fibered Heegaard Klein bottle invariant. For the prism manifold M(1,d)M(1,d) we construct the following groups of isometries S4S4, A5A5, and A4A4, which act on M(1,d)M(1,d) preserving the meridian fibering and not leaving any Heegaard Klein bottle invariant. Let G0G0 be the normal subgroup of G consisting of the isometries which leave every fiber invariant. We show that if G   acts on M(b,d)M(b,d) preserving the longitudinal fibering and not leaving any fibered Klein bottle invariant, then M(b,d)=M(b,2)M(b,d)=M(b,2) and G/G0G/G0 is isomorphic to one of the groups Z3Z3, Z6Z6, Dih(Z3)Dih(Z3) and Dih(Z6)Dih(Z6); and if G   preserves the meridian fibering not leaving any fibered Klein bottle invariant, then M(b,d)=M(1,d)M(b,d)=M(1,d) and G/G0G/G0 is isomorphic to one of the groups S4S4, A5A5, and A4A4. We give complete descriptions of the group G in each of these cases.