presentations of Sp(n, Z) and cocycles. 
Question: where can I find explicit presentations of the group $Sp(n, \mathbb Z)$, for small $n$?

It is known that $Sp(n, \mathbb Z)$ admits a $2$-cocycle $h$ with values in $\mathbb Z/2\mathbb Z$ which I'd like to view in the following way. If we fix a presentation with relations forming a set $R$, then $h$ is a function $R \to \mathbb Z/2\mathbb Z$ (which fulfils some condition).

Question: where can I find an explicit description of this cocycle, in the form of what values does it take on elements of some presentation?

 A: An explicit presentation was given by Behr in
MR0369562 (51 #5795) 
Behr, Helmut
Eine endliche Präsentation der symplektischen Gruppe Sp4(Z). (German) 
Math. Z. 141 (1975), 47–56. 
20H25 
And a slightly shorter one by P. Bender in 
Bender, Peter
Eine Präsentation der symplektischen Gruppe Sp(4,Z) mit 2 Erzeugenden und 8 definierenden Relationen. (German) 
J. Algebra 65 (1980), no. 2, 328–331. 
General $SP(2n, Z)$ can be reduced to $SP(4, Z),$ the argument is in
MR0133303 (24 #A3137) 
Klingen, Helmut
Charakterisierung der Siegelschen Modulgruppe durch ein endliches System definierender Relationen.
A: There is a simple and explicit presentation of $SP(2n,\mathbb{Z})$ in Theorem 9.2.13 of Hahn-O’Meara's book "The classical groups and K-theory".  In some sense, this presentation is related to the ideas of Klingen mentioned by Rivin; however, just following Klingen's method leads to an enormous presentation (something on the order of 5 families of generators and 67 families of relations), as is shown in the paper
MR0280606 (43 #6325) 
Birman, Joan S.
On Siegel's modular group. 
Math. Ann. 191 1971 59–68. 
Another simple presentation can be found in the paper
MR1152494 (93b:11054) 
Lu, Ning(1-RICE)
A simple presentation of the Siegel modular groups. 
Linear Algebra Appl. 166 (1992), 185–194. 
The basic idea is that there exist nice presentations for the mapping class group, which surjects onto the symplectic group.
A: For $Sp_4(\mathbf{Z})$, there is a presentation in H. Behr "Eine endliche Präsentation der symplektischen Gruppe $Sp_4(\mathbf{Z})$", Math. Z. 141 (1975), 47--56.
