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In the period 1994-1999 Thas wrote a series of three papers on generalized quadrangles of order (s, s^2). In this Part IV we classify all finite translation generalized quadrangles of order (s, s^2) having a kernel of size at least 3, containing a regular line not incident with the translation point. There are several applications on generalized quadrangles of order (s, s^2) having at least 2 translation points, on Moufang quadrangles, and concerning the theorem of Fong and Seitz classifying all groups with a BN-pair of rank 2.arXiv:2205.15260v1 fatcat:jgt6lcphmbgkbniblgrbz6gqey