Beta-1,3-N-acetylglucosaminyltransferase involved in the synthesis of poly-N-acetyllactosamine. Has activity for type 2 oligosaccharides. Also acts as a core1-1,3-N-acetylglucosaminyltransferase (Core1-beta3GlcNAcT) to form the 6-sulfo sialyl Lewis x on extended core1 O-glycans. 3-O-{beta-D-galactosyl-(1->3)-[N-acetyl-beta-D-glucosaminyl-(1->6)]-N-acetyl-alpha-D-galactosaminyl}-L-threonyl-[protein] + UDP-N-acetyl-alpha-D-glucosamine = 3-O-{beta-D-GlcNAc-(1->3)-beta-D-Gal-(1->3)-[beta-D-GlcNAc-(1->6)]-alpha-D-GalNAc}-L-threonyl-[protein] + H(+) + UDP 3-O-{beta-D-galactosyl-(1->3)-[N-acetyl-beta-D-glucosaminyl-(1->6)]-N-acetyl-alpha-D-galactosaminyl}-L-seryl-[protein] + UDP-N-acetyl-alpha-D-glucosamine = 3-O-{beta-D-GlcNAc-(1->3)-beta-D-Gal-(1->3)-[beta-D-GlcNAc-(1->6)]-alpha-D-GalNAc}-L-seryl-[protein] + H(+) + UDP a beta-D-galactosyl-(1->4)-N-acetyl-beta-D-glucosaminyl derivative + UDP-N-acetyl-alpha-D-glucosamine = an N-acetyl-beta-D-glucosaminyl-(1->3)-beta-D-galactosyl-(1->4)-N-acetyl-beta-D-glucosaminyl derivative + H(+) + UDP Protein modification; protein glycosylation. Belongs to the glycosyltransferase 31 family. beta 3 GlcNAc-T III