# What's the smallest permutation group which has a subgroup not known to be a Galois group of a polynomial over $\mathbb Q$?

The question of what is the group of smallest order for which the inverse Galois problem is open seems to attract more attention, for example, this MO thread. Apparently, the smallest order is $$672$$. However, by Cayley's theorem, a group of order $$n$$ embeds (in general) only into $$S_n$$. And $$S_{672}$$ is a truly massive group. On the other hand, according to Wikipedia, the Mathieu group $$M_{23}$$ has no known representation as a Galois group. $$M_{23}$$ is a subgroup of $$S_{23}$$, which has a more manageable $$23!\approx 2.5\cdot 10^{22}$$ elements, less than one mole.

So it is natural to ask: is there an $$n<23$$ such that $$S_n$$ has a subgroup for which the inverse Galois problem is unsolved? That is, for which it is not know if there's a polynomial in $$\mathbb Q[x]$$ which has that group as its Galois group.

And yet another framing: what's the smallest degree $$d$$ for which we do not know what the possible Galois groups of polynomials in $$\mathbb Q[x]$$ of degree $$d$$ are?

• $23!$ elements is still massive although the Monster group has much more elements. Oct 29, 2023 at 12:57
• The referenced MO thread seems to speak about not just any group of order $672$, but specifically about $PGL(2,7)$. Hence your embedding it in $S_{672}$ as if it were just a random group of same order seems a bit unfair Oct 29, 2023 at 13:08
• @HagenvonEitzen what I understood from Joachim's last comment is that the polynomial $x^8-x^7-29x^6+11x^5-139x^4+37x^3+32x^2-10x-1$ in Malle's paper has Galois group $PGL(2,7)$, and the group with smallest order for which the inverse Galois problem is unsolved is another group of order 672. Oct 29, 2023 at 13:23