With that SN, it would have to be a N-series (N 126169).
N 60,001 – N 190,000…...1972-74 (manufacture)
Only problem is that I don't know what .44 Special revolver was being made by S&W at that time. Is there a model number stamped in the yoke area? Would look like "MOD xx-x"
S126169 (the S and the number are separated on the butt) dates to sometime in 1955. The revolver could be a 1950 44 Military or if it has target sights, a 1950 44 Target revolver.
Measure the barrel again. Revolver barrels are measured from the front face of the cylinder to the muzzle. S&W's .44 Special barrels are usually 6-1/2 inches, rather than 6 inches.