Departamento de Matematica, Universidade de Aveiro, 3800 Aveiro, Portugal
Department of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom
Abstract: We classify the rotary hypermaps (sometimes called regular hypermaps) on an orientable surface of genus 2. There are 43 of them, of which 10 are maps (classified by Threlfall), 20 more can be obtained from the 10 maps by applying Machi's operations, and the remaining 13 may be obtained from the maps by using Walsh's bijection between maps and hypermaps. As a corollary, we deduce that there are no non-orientable reflexible hypermaps of characteristic $-1$.
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