$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
Variants have been asked here before (e.g. Which small finite simple groups are not yet known to be Galois groups over Q? specifically for simple groups), but I explicitly don't want to restrict to simple groups, since this let's one get to rather lofty orders all while dodging the true difficulty of solving embedding problems. Of course I'm aware that results in the literature are usually not custom-made to answer these kind of "trivia" questions, so here are some thoughts.
My own guess is that the answer may be as small as $\SL_2(13)$ (which has order $2184$), so that I'm essentially asking two types of sub-questions:
- Is there any quotable reference or reasonably short argument for the claim that all groups of order smaller than $2184$ have been realized? (Or alternatively, am I in fact missing some smaller open case?)
Since a counterexample would have to be (non-simple) nonsolvable involving a very small nonsolvable composition factor (concretely, $\PSL_2(q)$ with $q\le 11$), I guess I might be able to piece this together from various results in the literature about solvability of central embedding problems, split embedding problems, and embedding problems with (certain) nonabelian simple kernel, although I haven't checked in full and the full list of relevant groups is not that small after all. (A short argument might be of the form "every embedding problem is either of this or that type, and these types are known to be solvable when only the couple of smallest simple groups are involved.")
Update: After writing some code filtering the relevant nonsolvable groups, using some known general results such as solvability of split embedding problems with nilpotent kernel (Shafarevich), realizability of all central extensions of $A_n$ and $S_n$ and solvability of all embedding problems with kernel $A_n$ ($n\ne 6$) (see Malle-Matzat's book for the latter two), the orders I'm actually struggling to rule out directly are $16\cdot 60$, $32\cdot 60$, $8\cdot 168$, $4\cdot 360$ and $6\cdot 360$ (where in each case I've factored by the order of the relevant nonabelian composition factor), although in particular the last two cases might fall with a bit more effort.
- Is $\SL_2(13)$ indeed an open case? (And if not, what is the smallest open $\SL_2(p)$?)
The reason why $\SL_2(p)$ looks difficult is that, in order to solve the embedding problem from $\PSL_2(p)$, one necessarily needs to begin with a (suitable) totally real $\PSL_2(p)$-extension, which doesn't go too well with the "classical" realizations à la Shih from rigidity theory (although $p=7$ and $11$ have been solved). I'm not too familiar with the more recent realizations of $\PSL_2(p)$ (for all primes $p$) by Zywina, although I would have thought they don't yield totally real extensions either. In short, is there either a recent reference (say, from the last decade) stating that $p=13$ (or similar) is open, or otherwise does someone know a realization? (Just to make sure: the standard databases seem to know no such realization.)