Johnson Solids (Page 1)

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(J18) Elongated Triangular Cupola
Vertices:  15  (6[3] + {6 + 3}[4])
Faces:14  ({3 + 1} equilateral triangles + {3 * 3} squares
    + 1 regular hexagon)
Edges:27
Symmetry:  3-fold Pyramidal  (C3v)
(values below based on edge length = 1)
Volume:(5*sqrt(2)+9*sqrt(3))/6    ≈3.7765875133308951476