To formulate my question precisely: let $s\_k$ be the number of isomorphism types of automorphism groups of convex 3-dimensional polytopes with $k$ faces. Are there any references discussing the asymptotic growth of $(s\_k)$ as $k\rightarrow\infty$?



Growth of number of isomorphism types of automorphism groups of convex 3-dimensional polytopes
The possible symmetry groups of 3-dimensional polytopes have been complete classified (see [here](https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions)).


In particular, there are only a few families of symmetry groups with arbitrarily large size (according to [this section](https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions#The_seven_infinite_series_of_axial_groups) there are seven). And there are [seven other point groups](https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions#The_seven_remaining_point_groups) that do not generalize.


If you just care about assymptotics then we can do some very rough estiamtions.
I think it is safe to say that a 3-polytope with $k$ faces can have at most (but probably far less than) $100k$ symmetries.
Being optimistic, and assuming that with $k$ faces you can build a polytope
for each of the symmetry groups of size $\le 100k$, you will find (for sufficiently large $k$) that there are $\le 700k+7$ isomorphism types of groups.


This is a *very rough* estimate, but it gives you a **linear upper bound**.


