  We define geodesic normal forms for the general series of complex reflection
groups G(e,e,n). This requires the elaboration of a combinatorial technique in
order to explicitly determine minimal word representatives of the elements of
G(e,e,n) over the generating set of the presentation of Corran-Picantin. Using
these geodesic normal forms, we construct intervals in G(e,e,n) that are
lattices. This gives rise to interval Garside groups. We determine which of
these groups are isomorphic to the complex braid group B(e,e,n) and get a
complete classification. For the other Garside groups that appear in our
construction, we provide some of their properties and compute their second
integral homology groups in order to understand these new structures.
