Automorphism Groups of Compact Non-orientable Surfaces of Genus 6 with Extremal Metric Discs

We study a compact non-orientable Klein surface of genus 6 without boundary such that it contains a disc with the maximum radius determined by its genus. We discuss the group of automorphisms of such a surface. As a result, we find two surfaces that admit an automorphism of order 10, where 10 is the maximum order of automorphisms that compact non-orientable Klein surfaces of genus 6 without boundary can admit. Applying this result, we obtain compact non-orientable Klein surfaces of arbitrary even genus g≥4 admitting an automorphism of maximum order 2(g−1).