The special orthogonal group in dimension 32.
In type I string theory, and more generally in type II string theory on toroidal orientifolds, RR-field tadpole cancellation implies that the gauge group is SO(32) (see there).
Under the duality between type I and heterotic string theory this translates to the semi-spin gauge group SemiSpin(32) of heterotic string theory.
Discussion of type-I string phenomenology and grand unified theory based on SO(32) type-I strings: (MMRB 86, Ibanez-Munoz-Rigolin 98, Yamatsu 17).
rotation groups in low dimensions:
see also
Type I string phenomenology and discussion of GUTs based on SO(32) type I strings:
H.S. Mani, A. Mukherjee, R. Ramachandran, A.P. Balachandran, Embedding of $SU(5)$ GUT in $SO(32)$ superstring theories, Nuclear Physics B Volume 263, Issues 3–4, 27 January 1986, Pages 621-628 (doi:10.1016/0550-3213(86)90277-4)
Luis Ibáñez, C. Muñoz, S. Rigolin, Aspects of Type I String Phenomenology, Nucl.Phys. B553 (1999) 43-80 (arXiv:hep-ph/9812397)
Emilian Dudas, Theory and Phenomenology of Type I strings and M-theory, Class. Quant. Grav.17:R41-R116, 2000 (arXiv:hep-ph/0006190)
Naoki Yamatsu, String-Inspired Special Grand Unification, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 10, 1 (arXiv:1708.02078, doi:10.1093/ptep/ptx135)
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