Let \(G\) be a finite group. Denote by \(\pi(G)\) the set of prime divisors of the order of \(G\). The Gruenberg–Kegel graph (prime graph) of \(G\) is the graph with the vertex set \(\pi(G)\) in which two different vertices \(p\) and \(q\) are adjacent if and only if \(G\) has an element of order \(pq\). If \(|\pi(G)|=n\), then the group \(G\) is called \(n\)-primary. In 2011, A.S. Kondrat’ev and I.V. Khramtsov described finite almost simple \(4\)-primary groups with disconnected Gruenberg–Kegel graph. In the present paper, we describe finite almost simple \(4\)-primary groups with connected Gruenberg–Kegel graph. For each of these groups, its Gruenberg–Kegel graph is found. The results are presented in a table . According to the table, there are \(32\) such groups. The results are obtained with the use of the computer system GAP.

中文翻译：

---

有限的几乎简单的$$ 4 $$ 4-具有连通的Gruenberg–Kegel图的主要组

令 \（G \）为一个有限群。用\（\ pi（G）\）表示\（G \）阶素数集 。\（G \）的Gruenberg–Kegel图（素数图） 是顶点集为\（\ pi（G）\）的图，其中两个不同的顶点 \（p \）和 \（q \）相邻，如果并且仅当 \（G \）的元素顺序为 \（pq \）时。如果\（| \ pi（G）| = n \），则组 \（G \）称为\（n \）- primary。在2011年，AS Kondrat'ev和IV Khramtsov描述了有限的几乎简单的\（4 \）断开的Gruenberg–Kegel图的主要组。在本文中，我们用连通的Gruenberg–Kegel图描述了有限的几乎简单的\（4 \）-初等组。对于每个组，都可以找到其Gruenberg-Kegel图。结果列在表中。根据表，有\（32 \）个这样的组。使用计算机系统GAP获得结果。