With its 13 faces and its 13 vertices, this polyhedron has six 345vertices and a 3fold dihedral symmetry, but it is not minimal, neither for the number of faces nor for the number of vertices.
"G3" has also a 555vertex as has the regular dodecahedron; the first has a minimum of vertices and the second a minimum of faces. 
assembling of twenty cubes ring of eight regular dodecahedra 

ring of eight regular octahedra
(minimal deltahedron: 24 vertices and 48 faces) to get a ring of eight regular icosahedra

Johnson 18 drilled with a triangular cupola (Johnson 03) augmented with a triangular prism. truncated octahedron drilled 
references: 
• Adventures Among the Toroids by B.M. Stewart, 1970.
• http://www.orchidpalms.com/polyhedra/ (pages "acrohedra" and "toroids") by Jim McNeill 
January 2004 updated 30012008 