In this paper we study some aspects of the cohomology of groups and we construct a central extension of the symplectic group $S{p}_n\left(A\right)$.

All finite simple groups of Lie type of rank $n$ over a field of size $q$, with the possible exception of the Ree groups ${}^2{G}_2\left(q\right)$, have presentations with at most 49 relations and bit-length $O(\mathrm{𝚕𝚘𝚐}n+\mathrm{𝚕𝚘𝚐}q)$. Moreover, $A_n$ and $S_n$ have presentations with 3 generators; 7 relations and bit-length $O(\mathrm{𝚕𝚘𝚐}n)$, while $\mathrm{𝚂𝙻}(n,q)$ has a presentation with 6 generators, 25 relations and bit-length $O(\mathrm{𝚕𝚘𝚐}n+\mathrm{𝚕𝚘𝚐}q)$.