Johnson Solids (Page 1)

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(J19) Elongated Square Cupola
Vertices:  20  (8[3] + {8 + 4}[4])
Faces:18  (4 equilateral triangles + {4 * 3 + 1} squares
    + 1 regular octagon)
Edges:36
Symmetry:  4-fold Pyramidal  (C4v)
(values below based on edge length = 1)
Volume:(9+8*sqrt(2))/3    ≈6.77123616632825346347