Consider (e.g.) the full permutation group $G=S\_6$. A valid set of generators and equations for $G$ is $r^6=m^2=(rm)^5=1$. I say this system has width $3$ (because there are $3$ equations), length $10$ (because there are $10$ generators in $(rm)^5$ - arguably, $m$ as a mirror causes no "load" in which case the length would be $6$) and height $6$ (for the exponent $6$).  

What is the generator set with minimum height or length or width or (best) everything at the same time? How do I find it in the general case?



Generators for permutation groups
There is a well-known [Coxeter presentation](https://groupprops.subwiki.org/wiki/Presentation_theory_of_symmetric_groups) of $S\_n$ having height $3$.


Regarding presentations with short width and length (again, in the general case) you can have a look at the paper 


Bray, J. N.; Conder, M. D. E.; Leedham-Green, C. R.; O’Brien, E. A.: [*Short presentations for alternating and symmetric groups*](http://dx.doi.org/10.1090/S0002-9947-2011-05231-1), Trans. Am. Math. Soc. **363**, No. 6, 3277-3285 (2011), [ZBL1223.20023](https://zbmath.org/?q=an:1223.20023).


For instance, the authors show hot to construct a presentation in two generators having width $O(\log n)$ and length $O(\log^2 n)$.


