To make this first symmetrohedron, I started with the rhombicosidodecahedron, augmented its thirty square faces with antiprisms, and then formed the convex hull of that solid. (I did all of this using Stella 34d: Polyhedron Navigator, which you can try for free right here.) The resulting polyhedron contains, as faces, twelve regular pentagons, twenty equilateral triangles, thirty squares, and sixty isosceles trapezoids, or 122 faces in all.

Next, I applied Stella’s “try to make faces regular” function, which produced the solid seen below. This had the effect of transforming the squares into rhombi, and changing the trapezoids so that, while still isosceles trapezoids, they closely resemble squares.