Hypermap operations of finite order

Duality and chirality are examples of operations of order 22 on hypermaps. James showed that the groups of all operations on hypermaps and on oriented hypermaps can be identified with the outer automorphism groups OutΔ≅PGL2(Z) and OutΔ+≅GL2(Z) of the groups Δ=C2∗C2∗C2Δ=C2∗C2∗C2 and Δ+=F2Δ+=F2. We will consider the elements of finite order in these two groups, and the operations they induce.