An orthogonal group is any member of a sequence of classical orthogonal matrix groups.

For every the orthogonal group for dimension is the subgroup of the general linear group given by all orthogonal -matrices with complex entries, i.e., the set

where, if , then is the complex conjugate of and the transpose and where is the identity -matrix.

The orthogonal groups are an easy family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a category of partitions. More precisely, it is the category of all pair partitions that induces the corepresentation categories of . Its canonical generating partition is .