Symmetrohedra are symmetric polyhedra with many faces regular, but not necessarily all of them. The symmetrohedron shown above is the dual of the convex hull of the compound of the great rhombicosidodecahedron and its dual, the disdyakis triacontahedron. All of its faces are regular, except for the triangles, which are scalene.
The second symmetrohedron shown here is the dual of the convex hull of the great rhombcuboctahedron and its dual, the disdyakis dodecahedron. Like its “big brother” above, all of this symmetrohedron’s faces are regular, except for the scalene triangles.
These polyhedra were made using Stella 4d, a program you can try for free at this website.
RobertLovesPi.net 
Correction; The convex core!
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Until I saw this comment, and played around with Stella some more, I did not realize that “form convex hull” and “form convex core” were reciprocal operations — so the dual of the convex hull is the convex core. Clearly, I have some studying to do.
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