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The gyroelongated pentagonal pyramid is Johnson solid 
and can be obtained by cumulating the top of a pentagonal
antiprism, or alternately by replacing one pentagonal
pyramid of an icosahedron with a pentagon.
Removing a second pentagonal pyramid from an
icosahedron gives a metabidiminished
icosahedron.
The skeleton of the gyroelongated pentagonal pyramid 
appeared in Zaks (1976) and was used by Owens (1980) in the construction of a 76-node
polyhedral quintic
nonhamiltonian graph (though neither author identified the graph as the skeleton
of any particular polyhedron).
