Projective (or spin) representations of finite groups. III

In previous papers I and II under the same title, we proposed a practical method called Efficient stairway up to the Sky, and apply it to some typical finite groups G, with Schur multiplier M (G) containing prime number 3, to construct explicitly their representation groups R (G), and then, to construct a complete set of representatives of linear IRs of R (G), which gives naturally, through sectional restrictions, a complete set of representatives of spin IRs of G. In the present paper, we are concerned mainly with group G=G₃₉ of order 27 in a list of Tahara's paper, with M (G) = Z₃ Z₃. In this case, to arrive up to the Sky, we have two steps of one-step efficient central extensions. By the 1st step, we obtain a covering group of order 81, and by the 2nd step we arrive to R (G) of order 243. At the 1st step, to construct explicitly a complete set of representatives of IRs of the group, we apply Mackey's induced representations, and at the 2nd step, with a help of this result, we apply so-called classical method for semidirect product groups given by Hirai and arrive to a complete list of IRs of R (G). Then, using explicit realization of these IRs, we can compute their characters (called spin characters).