Histone methyltransferase that specifically trimethylates 'Lys-9' of histone H3. H3 'Lys-9' trimethylation represents a specific tag for epigenetic transcriptional repression by recruiting HP1 (CBX1, CBX3 and/or CBX5) proteins to methylated histones. Mainly functions in euchromatin regions, thereby playing a central role in the silencing of euchromatic genes. H3 'Lys-9' trimethylation is coordinated with DNA methylation. Plays a role in promoter hypermethylation and transcriptional silencing of tumor suppressor genes (TSGs) or other tumor-related genes. Also required to maintain a transcriptionally repressive state of genes in undifferentiated embryonic stem cells (ESCs). Associates at promoter regions of tumor suppressor genes (TSGs) leading to their gene silencing (By similarity). L-lysyl(4)-[histone H3] + 3 S-adenosyl-L-methionine = 3 H(+) + N(6),N(6),N(6)-trimethyl-L-lysyl(4)-[histone H3] + 3 S-adenosyl-L-homocysteine Associated with non-pericentromeric regions of chromatin. Excluded from nucleoli and islands of condensed chromatin (By similarity). 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. Histone-lysine methyltransferase family. Suvar3-9 subfamily.