<p>In this paper, we are concerned with the compactness and the explicit expression for the automorphism group, as well as the rigidity of proper holomorphic self-mappings of a class of unbounded pseudoconvex domains with an exponentially flat boundary point, denoted by the Greene-Krantz-Hartogs domains. As a primary contribution, we show, on the one hand, that the automorphism group of this domain is compact and then determine its full expression. On the other hand, we derive rigidity results for proper holomorphic self-mappings of the Greene-Krantz-Hartogs domains without any dimensional restriction.</p>