Histone methyltransferase that specifically trimethylates 'Lys-20' of histone H4 to form trimethylated histone H4 lysine 20 (H4K20me3) which represents a specific tag for epigenetic transcriptional repression. L-lysyl(20)-[histone H4] + 3 S-adenosyl-L-methionine = 3 H(+) + N(6),N(6),N(6)-trimethyl-L-lysyl(20)-[histone H4] + 3 S-adenosyl-L-homocysteine Belongs to the class V-like SAM-binding methyltransferase superfamily.