Duality and chirality are examples of operations of order 2 on hypermaps.
James showed that the groups of all operations on hypermaps and on oriented
hypermaps can be identified with the outer automorphism groups Out ∼=
PGL2(Z) and Out + ∼=
GL2(Z) of the groups = C2 ∗C2 ∗C2 and + = F2. We
will consider the elements of finite order in these two groups, and the operations
they induce