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?