Johnson Solids (Page 1)

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(J10) Gyroelongated Square Pyramid
Vertices:  9  ({4 + 1}[4] + 4[5])
Faces:13  ({4 * 3} equilateral triangles + 1 square)
Edges:20
Symmetry:  4-fold Pyramidal  (C4v)
(values below based on edge length = 1)
Volume:(sqrt(2)+2*sqrt(4+3*sqrt(2)))/6    ≈1.1927022422322325591