Hypermap operations of finite order

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. MSC classification: Primary 05C10, secondary 05C25, 20F28.