OK just one more
It is a 3 speed "Ford-O-Matic" just like GBird stated. There were two "versions" of it in 56. Early 56 Birds had an Air Cooled left over from the 55s. Later had lines running to the radiator and were cooled that way.
The Ford-O-Matic name was used twice to mean two different transmissions. Yea, confusing. Here is a quick and dirty article that gives you some history and about the name change to the 2-Speed.
Cruise-O-Matic - Wikipedia, the free encyclopedia
HTH