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).