in $G$ and by $C_i$ the $\a$-regular conjugacy class of $G$ containing $g_i,1\leq i\leq r$. Then the elements $$z_i=\sum_{t\in T_i}\bar t \overline{g_i}\overline{t}^{-1}\ \ \ (1\leq i\leq r)$$ constitute an $F$-basis $Z(F^{\a}[G])$. In particular, if $\a$ is a normal cocycle, then the elements $$z_i=\sum_{g\in C_i}\bar g\ \ \ (1\leq i\leq r)$$ constitute an $F$-basis $Z(F^{\a}[G])$.

The Generalized twisted double {#s3}
==============================

In this section we construct a finite dimensional semisimple associative algebra $A_{\alpha}(G,S)$ over $\C$ associated to a finite group $G,$ a finite right $G$-set $\S$ and a suitable 2-cocycle $\alpha$. Although this construction in the present form is influenced by the work in \[DPR\], \[M\] and \[DM5\] and Section 6 of this paper, its origin goes back to the earlier work of \[S\], \[BCM\] and \[DT\] in Hopf algebra, where it is called crossed product. In the case the $G$-set $\S$ consists of a dual basis of $\C[G]^*,$ $A_{\alpha}(G,\S)$ is exactly the twisted double studied in \[M\] and \[DM5\]. This should explain why we call $A_{\alpha}(G,\S)$ a generalized twisted double. The exposition of this section follows \[M\] closely and ideas are also similar.

Let $G$ be any finite group with identity $1$ and $\S$ be a finite right $G$-set. Set 