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](https://www.gap-system.org/Packages/smallgrp.html) catalogizes groups of small order,
* The [GAP Transitive Groups Library](https://www.gap-system.org/Packages/transgrp.html) and the [GAP Primitive Groups Library](https://www.gap-system.org/Packages/primgrp.html)
catalogize transitive, respectively primitive, subgroups of symmetric groups
of small degree,
* The [GAP Perfect Groups Library](https://www.gap-system.org/Datalib/perfect.html) 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](https://mathoverflow.net/questions/15094/) 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.



Catalogue of groups with short finite presentations
I would very much like to have such a database and would like to contribute to its development. Prompted by this question, we talked about what such a database could look like (e.g. in terms of groups covered, functionality etc.) at a discussion session of a workshop in Manchester with Ian Leary, Marco Linton, Saul Schleimer and Henry Wilton. I encourage anyone interested in this to contact me.


As some inspiration, I'll link this [mathoverflow question on atlas-like websites](https://mathoverflow.net/questions/354327).


