  In this note we investigate three new pencils of symmetric surfaces in
complex projective three-space. These have degree 6, 8 resp. 12 and are
invariant under the action of subgroups of SO(4) containing the Heisenberg
group. The pencils of degree 6 and 12 are invariant under the action of bigger
groups, precisely under the action of the reflection groups F_4 resp. H_4. We
compute equations for the generators of the pencils and describe the base locus
of each pencil. We find also the singular surfaces and their number of
singularities, which are, in fact, ordinary double points. In degree 12, we get
a surface with 600 nodes. This confirms a conjecture of V. Goryounov and
presents a new lower bound for the maximal number of nodes of such a surface.
