In Symmetries of partial Latin squares, it is shown that for any graph $\Gamma=(V,E)$ with automorphism group $G$, there is a partial Latin square with $|V|+3|E|+49$ filled cells whose autotopism group is isomorphic to $G$.
Hence my question...
Which graph with an automorphism group isomorphic to the quaternion group $Q_8$ minimizes $|V|+3|E|$?
The answer to the question Smallest graph with automorphism group the quaternion $8$-group, $Q_8$ gives an example of a 16-vertex 48-edge graph with automorphism group $Q_8$. It could be one that minimizes $|V|+3|E|$.
I'm looking to find partial Latin rectangles that (a) have symmetry groups isomorphic to a given group $G$, and (b) have low weight [number of filled cells]. The quaternion group is among the groups of smallest order I have yet to compute examples. The answer to this question will give an upper bound on the weight.