n=30 Icosidodecahedron. Images: A critical points scheme for Generating uniformly distributed points on a sphere.
The applet illustrates the case where 30 vertices of a Icosidodecahedron "induce" the vertices of two other polyhedra:
V=32 ●Pentakis Dodecahedron← V=30 ●Icosidodecahedron →V=60 ☐Rhombicosidodecahedron.
Generating polyhedra is in https://www.geogebra.org/m/hymcebuw. Description are in https://www.geogebra.org/m/y8dnkeuu and https://www.geogebra.org/m/rkpxwceh.