I considered the cominational problem of finding out how many ways there are to augment a given polytope where rotations and reflections are regarded as one (put pyramids or something similar on each facet) but I still need to identify how much info about the shape is needed: isometries, facet count, facet of facet count etc,
for 2D it's straight forward given that the chiral pairs (where the polygon can be flipped over) and rotations are classed as the samem the sequence for a n-gon is as follows:
where the degenerate monogon and digon form the first two terms and when the shape hasn't been augemented that's excluded
It is also very easy to identify the isometries of a 2D shape, for higher dimensions find a element at which it is transitive and you just need it's isometries (number it has itself times the figure's isometries) it doesn't have to be all of teh elements of a given dimension (just the same type) I'm guessing that for when you get more than one answer for isometries then it's the smallest one (I haven't done much investiagation into isometry yet)
The problem can be interesting because it's not as simple as taking the number of permutations (which is usually easy (n! where n is the number of facets, at least for a facet transitive polytope)) and then dividing by the number of isometries:
Why did the chicken cross the Mobius Band?
To get to the other ...um...!!!