  Many critical properties of the Hintermann-Merlini model are known exactly
through the mapping to the eight-vertex model. Wu [J. Phys. C {\bf 8}, 2262
(1975)] calculated the spontaneous magnetizations of the model on two
sublattices by relating them to the conjectured spontaneous magnetization and
polarization of the eight-vertex model, respectively. The latter conjecture
remains unproved. In this paper, we numerically study the critical properties
of the model by means of a finite-size scaling analysis based on transfer
matrix calculations and Monte Carlo simulations. All analytic predictions for
the model are confirmed by our numerical results. The central charge $c=1$ is
found for the critical manifold investigated. In addition, some unpredicted
geometry properties of the model are studied. Fractal dimensions of the largest
Ising clusters on two sublattices are determined. The fractal dimension of the
largest Ising cluster on the sublattice A takes fixed value $D_{\rm
a}=1.888(2)$, while that for sublattice B varies continuously with the
parameters of the model.
