The following arose as a side issue in a project on graph reconstruction.

**Problem:** Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a good upper bound on $a(n)$.

There is a paper of Opstall and Veliche that finds the maximum over all cubic graphs, but the maximum occurs for graphs very far from being 3-connected.

I foolishly **conjecture**: for $n\ge 16$, $a(n) < n 2^{n/4}$.

When $n$ is a multiple of 4 there is a vertex-transitive cubic graph achieving half the conjectured bound, so if true the bound is pretty sharp.

The maximum does not always occur for vertex-transitive graphs. There is no vertex-transitive counterexample to the conjecture with less than 1000 vertices.

Any bound for which the exponential part is $2^{cn}$ for $c<\frac12$ is potentially useful.

**Added:** As verret pointed out in a comment, a paper of Potočnik, Spiga and Verret, together with the computation I mentioned, establishes the conjecture for vertex-transitive graphs, so the remaining problem is whether one can do better for non-transitive graphs. For 20, and all odd multiples of 2 vertices from 18 to at least 998 (but not for 4-16 or 24 vertices) the graph achieving the maximum is not vertex-transitive.