  A geometric analysis of the $sdg$ interacting boson model is performed. A
coherent-state is used in terms of three types of deformation: axial quadrupole
($\beta_2$), axial hexadecapole ($\beta_4$) and triaxial ($\gamma_2$). The
phase-transitional structure is established for a schematic $sdg$ hamiltonian
which is intermediate between four dynamical symmetries of U(15), namely the
spherical ${\rm U}(5)\otimes{\rm U}(9)$, the (prolate and oblate) deformed
${\rm SU}_\pm(3)$ and the $\gamma_2$-soft SO(15) limits. For realistic choices
of the hamiltonian parameters the resulting phase diagram has properties close
to what is obtained in the $sd$ version of the model and, in particular, no
transition towards a stable triaxial shape is found.
