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 Δ ≅ P G L 2 ( Z ) and Out Δ + ≅ G L 2 ( Z ) of the groups Δ = C 2 ∗ C 2 ∗ C 2 and Δ + = F 2 . We will consider the elements of finite order in these two groups, and the operations they induce.