The Moebius-Kantor graph MK=G(8,3) is a Cayley graph of three non-abelian groups, the Pauli group P(1), the semi-dihedral group SD(16), as well as the dihedral group D(16) of order 16. In topological graph theory, it illustrates the Heawood number 7 of the torus and leads to the Tucker group Aut(MK), the unique group of genus 2. We compute the Lefschetz numbers to illustrate the Brouwer-Lefschetz fixed point theorem. MK is also the dual of the 2-skeleton complex of the 3-sphere G. The graph represents one of flat Clifford tori of a Hopf fibration in the 3-sphere G=K(2,2,2,2) reflecting that Coxeter saw that MK is a subgraph of the tesseract G*. It carries a metric d so that (MK,d) has only one algebraic group structure (P(1),*) that preserves the metric. It makes the Pauli group natural, similarly as the Moebius ladder M(16) makes the dihedral group D(16) natural, forcing the algebraic structure from the metric structure.

翻译：莫比乌斯–坎托图 MK=G(8,3) 是三个非阿贝尔群——即泡利群 P(1)、半二面体群 SD(16) 以及阶为 16 的二面体群 D(16)——的凯莱图。在拓扑图论中，它诠释了环面的希伍德数 7，并导出了图的自同构群 Aut(MK)，这是唯一一个亏格为 2 的群。我们计算了莱夫谢茨数，以阐释布劳威尔–莱夫谢茨不动点定理。MK 同时也是三维球面 G 的 2-骨架复形的对偶图。该图表示三维球面 G=K(2,2,2,2) 中一次霍普夫纤维化的扁平克利福德环面之一，反映了考克斯特曾指出的 MK 是超立方体图 G* 的一个子图这一事实。它承载一个度量 d，使得 (MK,d) 仅有一种保持度量的代数群结构 (P(1),*)。这赋予了泡利群以自然性，类似于莫比乌斯梯 M(16) 赋予二面体群 D(16) 以自然性，从而迫使代数结构由度量结构所决定。