Projective (or spin) representations of finite groups. I

Schur multiplier $M(G)$ of a finite group G has been studied heavily. To proceed further to the study of projective (or spin) representations of G and their characters (called spin characters), it is necessary to construct explicitly a representation group $R(G)$ of G, a certain central extension of G by $M(G)$, since projective representations of G correspond bijectively to linear representations of $R(G)$. We propose here a practical method to construct $R(G)$ by repetition of one-step efficient central extensions according to a certain choice of a series of elements of $M(G)$. This method is also helpful for constructing linear representations of $R(G)$ and accordingly for calculating spin characters. Actually, we will apply this method to several examples of G with prime number 3 in $M(G)$.