  The dynamical system on T^2 which is a group extension over an irrational
rotation on T^1 is investigated. The criterion when the extension is minimal, a
system of order 2 and when the maximal equicontinuous factor is the irrational
rotation is given. The topological complexity of the extension is computed, and
a negative answer to the latter part of an open question raised by
Host-Kra-Maass is obtained.
