Histone methyltransferase involved in left-right axis specification in early development and mitosis. Specifically trimethylates 'Lys-9' of histone H3 (H3K9me3). H3K9me3 is a specific tag for epigenetic transcriptional repression that recruits HP1 (CBX1, CBX3 and/or CBX5) proteins to methylated histones. Contributes to H3K9me3 in both the interspersed repetitive elements and centromere-associated repeats. Plays a role in chromosome condensation and segregation during mitosis (By similarity). N(6),N(6)-dimethyl-L-lysyl(9)-[histone H3] + S-adenosyl-L-methionine = H(+) + N(6),N(6),N(6)-trimethyl-L-lysyl(9)-[histone H3] + S-adenosyl-L-homocysteine In the pre-SET domain, Cys residues bind 3 zinc ions that are arranged in a triangular cluster; some of these Cys residues contribute to the binding of two zinc ions within the cluster. Belongs to the class V-like SAM-binding methyltransferase superfamily.