A symmetric group is any member of a certain sequence of classical orthogonal matrix groups.

For every the symmetric group for dimension is the subgroup of the general linear group given by all permutation -matrices, i.e., the set

where .

The symmetric 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. In fact, the category of all partitions induces the corepresentation categories of . It is canonically generated by .