Johnson Solids (Page 2)

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(J31) Pentagonal Gyrobicupola
Vertices:  20  ({10 * 2}[4])
Faces:22  (10 equilateral triangles + 10 squares + 2 regular pentagons)
Edges:40
Symmetry:  5-fold Antiprismatic  (D5v)
(values below based on edge length = 1)
Volume:(5+4*sqrt(5))/3    ≈4.6480906366663862619