In spin geometry, a spin<sup>h</sup> group (or quaternionic spin group) is a Lie group obtained by the spin group through twisting with the first symplectic group. H stands for the quaternions, which are denoted . An important application of spin<sup>h</sup> groups is for spin<sup>h</sup> structures.
The spin group is a double cover of the special orthogonal group , hence acts on it with . Furthermore, also acts on the first symplectic group through the antipodal identification . The spin<sup>h</sup> group is then:
mit . It is also denoted . Using the exceptional isomorphism , one also has with:
For all higher abelian homotopy groups, one has:
for .