We examine the superradiance of a Bose-Einstein condensate pumped with a Laguerre-Gaussian laser of high winding number, e.g., l = 7. The laser beam transfers its orbital angular momentum (OAM) to the condensate at once due to the collectivity of the superradiance. An l-fold rotational symmetric structure emerges with the rotatory superradiance. l number of single-charge vortices appear at the arms of this structure. Even though the pump and the condensate profiles initially have cylindrical symmetry, we observe that it is broken to l-fold rotational symmetry at the superradiance. Breaking of the cylindrical symmetry into the l-fold symmetry and OAM transfer to the condensate become significant after the same critical pump strength. Reorganization of the condensate resembles the ordering in the experiment by Esslinger and colleagues (2010 Nature 264 1301). We numerically verify that the critical point for the onset of the reorganization, as well as the properties of the emitted pulse, conform to the characteristics of superradiant quantum phase transition.