This is the snub cube. It’s one of the thirteen Archimedean solids.
The first modification I made to this polyhedron was to stellate it once.
The next step was to augment each yellow face with a tall prism.
Next, I formed the convex hull of the above solid.
The software I use to manipulate polyhedra, Stella 4d, has a function called “try to make faces regular.” The last step of this short polyhedral journey was to use this function on the above convex hull.
To make this polyhedron, I started with a snub cube. Next, I augmented all triangular faces of it with prisms, then took the convex hull of the result. Finally, I used Stella 4d‘s “try to make faces regular” function on the convex hull.
Stella 4d: Polyhedron Navigator has a free trial download available here.
This cluster-polyhedron was made with Stella 4d, software you can try at this website. Above, it is colored by face-type, referring to each face’s position within the overall cluster. In the image below, the original compound of five cubes contained one cube each, of five colors, and then each snub cube “inherited” its color from the cube to which it was attached.
In the next version, the colors are chosen by the number of sides of each face.