Johnson Solids (Page 2)

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(J23) Gyroelongated Square Cupola
Vertices:  20  ({4 * 3}[4] + 8[5])
Faces:26  ({8 + 4 * 3} equilateral triangles + {4 + 1} squares
    + 1 regular octagon)
Edges:44
Symmetry:  4-fold Pyramidal  (C4v)
(values below based on edge length = 1)
Volume:(3+2*sqrt(2)+2*sqrt(2*(2+sqrt(2)    
    +sqrt(146+103*sqrt(2)))))/3    
≈6.2107657920392000367