Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, and the question whether there exists a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order $2^s$ for every $s⩾3$ has been wide open. This question is answered in the affirmative in this paper via the construction of a connected tetravalent half-arc-transitive graph with vertex stabilizer ${\mathrm{D}}_8^2\times {\mathrm{C}}_2^m$ for each integer $m⩾1$, where ${\mathrm{D}}_8^2$ is the direct product of two copies of the dihedral group of order 8 and ${\mathrm{C}}_2^m$ is the direct product of m copies of the cyclic group of order 2. The graphs constructed have surprisingly many significant properties in various contexts.

半弧传递图是连接图论，黎曼曲面和群论的一个引人入胜的话题。尽管在过去的半个世纪中取得了丰硕的成果，但使用规定的顶点稳定器构造半弧传递图仍然具有挑战性。直到最近，只有六个已知的带有非阿贝尔顶点稳定剂的连接的四价半弧-传递图，以及是否存在一个带非阿贝尔顶点稳定剂的连接的四价半弧-传递图的问题$2^s$ 每一个 $s⩾3$已经敞开了。通过建立带有顶点稳定器的连接的四价半弧-传递图的连接，该问题得到了肯定的回答。${\mathrm{d}}_8^2\times {\mathrm{C}}_2^{米}$ 对于每个整数 $米⩾\mathrm{1个}$，在哪里 ${\mathrm{d}}_8^2$ 是8阶二面体组的两个副本的直接乘积， ${\mathrm{C}}_2^{米}$是2阶环状基团的m个副本的直接乘积。在各种情况下，构造的图都具有令人惊讶的许多重要特性。