Serial numbers ran from S150000 in 1956 to S175999 in 1957.
S&W started stamping model numbers at SN 179000 in 1957.
Yours would be a pre model 29 or simply "44 Magnum".
Your barrel length is likey 6.5", measured from the face of the cylinder to the end of the muzzle.
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The serial number is K 846xx
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It's a K-38, also known as the Target Masterpiece. SN dates it to 1950. SN's ran from K84150 to K104047 that year.
It went on to become the model 14 when S&W began assigning model numbers in 1957.
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S&W model 19-3 serial number 9K64xxx
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1976.
SN's ran from 9K10001 to 9K99999 that year.
Jim