For various types of groups, there exist catalogues of those groups of the particular type which are "small" in a certain sense. — For example:
The GAP Small Groups Library catalogizes groups of small order,
The GAP Transitive Groups Library and the GAP Primitive Groups Library catalogize transitive, respectively primitive, subgroups of symmetric groups of small degree,
The GAP Perfect Groups Library catalogizes perfect groups of small order,
and so on.
Question: Does there exist such data library for groups with "short" finite presentations?
Remarks:
There is a 12-years-old question asking for such database. At that time, apparently such database was not available. But maybe this has changed in the meantime(?)
When setting the length of a finite presentation equal to the sum of the lengths of the relators, for every presentation on two generators of length at most $10$, it is straightforward to decide whether the corresponding group $G$ is trivial, nontrivial but finite, or infinite (if $G$ is infinite, this can be seen from the abelian invariants of $G$, $G'$ or $G''$, and if $G$ is finite, coset enumeration finishes even with very small limits). Where things start to get more interesting is length $11$. One of the few examples for which deciding finiteness seems more tricky is the group $$ G \ := \ \langle a, b \ | \ a^3 = ab^{-3}a^{-1}b^{-1}a^{-1}b = 1 \rangle. $$ What is easy to see is that $G/G'' \cong {\rm C}_{37} \rtimes {\rm C}_9$, and that $G''$ is perfect — but beyond that, things seem to get more difficult. Almost for sure, people have considered this presentation (and other similar presentations) before. Looking into an appropriate data library would tell what is known about that (and other similar) groups immediately.
