We consider Riemann surfaces of even genus g with the action of the group \(\mathcal {D}_n\times \mathbb {Z}_2\), with n even. These surfaces have the maximal number of 4 non-conjugate symmetries and shall be called s-extremal. We show various results for such surfaces, concerning the total number of ovals, topological types of symmetries, hyperellipticity degree and the minimal genus problem. If in addition an s-extremal Riemann surface has the maximal total number of ovals, then it shall simply be called extremal. In the main result of the paper we find all the families of extremal Riemann surfaces of even genera, depending on if one of the symmetries is fixed-point free or not.

我们考虑在组\（\ mathcal {D} _n \ times \ mathbb {Z} _2 \）的作用下偶数为g的黎曼曲面，其中n个偶数。这些表面具有4个非共轭对称的最大数目，应称为s-extremal。我们显示了此类表面的各种结果，涉及椭圆的总数，对称的拓扑类型，超椭圆度和最小类问题。此外，如果s极值Riemann曲面具有最大的椭圆总数，则应简称为极值。在本文的主要结果中，我们找到了均匀属的所有极端Riemann曲面族，这取决于对称性之一是否是无定点的。