  In this paper, we describe the structural properties of the cone of
$\mathcal{Z}$-transformations on the second order cone in terms of the
semidefinite cone and copositive/completely positive cones induced by the
second order cone and its boundary. In particular, we describe its dual as a
slice of the semidefinite cone as well as a slice of the completely positive
cone of the second order cone. This provides an example of an instance where a
conic linear program on a completely positive cone is reduced to a problem on
the semidefinite cone.
