DCT: Dual clutch gearbox is referred to as DCT for short. Because it has two sets of clutches, some people call it "powershift".
the difference between p>DSG and DCT: the electronic module for shifting gears is different. The 6-speed one is from Borgwarner, USA, and it is called DCT gearbox in Borgwarner. Later, all vehicle manufacturers called DCT gearbox because of the need of publicity.
and DSG is the name of German Volkswagen, because Volkswagen also has a 7-speed gearbox. In Volkswagen, the 7-speed DSG is called DQ2, while the 6-speed dsg is called DQ25, and DQ25 is the 6-speed DCT of Borg Warner.
so to sum up, DSG generally refers to Volkswagen's 7-speed, while DCT generally refers to all 6-speeds.