\,\theta _{y,z} ),\qquad 
     x,y,z\in G,$$ for each triple in $G$. Recall the simplicial model $$\label{eq:3}
     \xymatrix@1{\textnormal{$*$} \;\;&\;\; G\ar@<.5ex>[l] \ar@<-.5ex>[l]
     \;\;&\;\; G^2\ar@<1ex>[l] \ar[l] \ar@<-1ex>[l] \;\;&\;\; 
      G^3 \ar@<1.5ex>[l] \ar@<.5ex>[l] \ar@<-.5ex>[l]
     \ar@<-1.5ex>[l] \;\;&\;\; 
      G^4 \ar@<2ex>[l] \ar@<1ex>[l] \ar[l] \ar@<-1ex>[l]
     \ar@<-2ex>[l] \;\;\cdots}$$ for $BG$, which is quite useful for computing cohomology [@B]. Write $S_p=G^p$ for the $p^{\textnormal{th}}$ space in . In these terms a central extension is given by a line bundle ${K^\tau }\to S_1$, a trivialization  of the alternating tensor product of its pullback to $S_2$, the latter constrained so that the alternating product of its pullbacks to $S_3$ is trivial . A proof of Proposition \[thm:6\] may be found in [@AS Proposition 6.3], [@FHT Part 1,§2.2.1], [@Bry Appendix].

Recall that the complex group algebra ${{{\mathbb C}}[G]}$ of $G$ is the abelian group of complex functions on $G$ with multiplication given by convolution with respect to Haar measure, that is, pushforward under multiplication $m{\colon}G\times G\to G$. If $\dim G>0$ then we take the continuous functions on $G$, which form a topological ring using the sup norm. If $\dim G=0$ ($G$ is finite) then the center of ${{{\mathbb C}}[G]}$ is the commutative algebra with natural basis the $\delta $-functions on the conjugacy classes. Alternatively, it 