Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

We study a form of refined class number formula (resp. type formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups right ideal classes unit orders). For each finite noncyclic group $G$, we give an explicit conjugacy whose modulo center are isomorphic to and write down representative class. This leads complete recipe (even formulas special cases) all groups. As application, prove existence superspecial abelian surfaces endomorphism coincide with $\mathbb{Q}(\sqrt{p})$ positive characteristic $p\not\equiv 1\pmod{24}$.