TITLE: A polytope with congruent facets and an insphere that is not facet-transitive?
QUESTION [6 upvotes]: Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties?

All facets are congruent, 
it has an insphere (a sphere to which each facet is tangent to), and
it is not facet-transitive.

In 3-dimensional space there is an example with the "memorable" name Pseudo-deltoidal icositetrahedron, depicted below.
I believe its the only such polyhedron.
I am not aware of any higher dimensional examples.

REPLY [4 votes]: First, a simple remark: If a polytope with congruent facets is inscribed in a sphere, then it is circumscribed about a sphere as well, and the two spheres are concentric.
Next, there is a series of examples described and pictured in my old question
Can the sphere be partitioned into small congruent cells? .  Each of these examples is what you want in $R^3$. If you begin with any one such example and place it on a great 2-sphere of the 3-sphere in $R^4$ (say, the "equator"), then suspend it from the poles, you will get an example answering your question. The construction generalizes inductively to all higher dimensions.