We discuss new-type grand unified theories based on grand unified groups broken to their special subgroups as well as their regular subgroups. In the framework, when we construct four-dimensional (4D) chiral gauge theories, i.e., the Standard Model (SM), 4D gauge anomaly cancellation restricts the minimal number of generations of the 4D SM Weyl fermions. We show that in a six-dimensional (6D) $SU(16)$ gauge theory on $M^4\times T^2/\mathbb{Z}_2$ one generation of the SM fermions can be embedded into a 6D bulk Weyl fermion. For the model including three chiral generations of the SM fermions, the 6D and 4D gauge anomalies on the bulk and fixed points are canceled out without exotic 4D chiral fermions.

9 pages, 1 table, no figures