In this paper, we first study biplanes $\mathcal{D}$ with parameters $(v,k,2)$, where the block size $k\in\{13,16\}$. These are the smallest parameter values for which a classification is not available. We show that if $k=13$, then either $\mathcal{D}$ is the Aschbacher biplane or its dual, or $Aut(\mathcal{D})$ is a subgroup of the cyclic group of order $3$. In the case where $k=16$, we prove that $|Aut(\mathcal{D})|$ divides $2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13$. We also provide an example of a biplane with parameters $(16,6,2)$ with a flag-transitive and point-primitive subgroup of automorphisms preserving a homogeneous cartesian decomposition. This motivated us to study biplanes with point-primitive automorphism groups preserving a cartesian decomposition. We prove that such an automorphism group is either of affine type (as in the example), or twisted wreath type.

中文翻译：

---

双翼飞机的对称性

在本文中，我们首先研究带有参数 $(v,k,2)$ 的双平面 $\mathcal{D}$，其中块大小 $k\in\{13,16\}$。这些是无法进行分类的最小参数值。我们证明如果 $k=13$，那么 $\mathcal{D}$ 是 Aschbacher 双平面或其对偶，或者 $Aut(\mathcal{D})$ 是 $3$ 阶循环群的子群。在$k=16$的情况下，我们证明$|Aut(\mathcal{D})|$除以$2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13 $. 我们还提供了一个带有参数 $(16,6,2)$ 的双平面示例，该双平面带有标志传递和点本原自同构子群，保留齐次笛卡尔分解。这促使我们研究具有保留笛卡尔分解的点原始自同构群的双平面。