On a group extension involving the sporadic Janko group $J_{2}$

According to the electronic Atlas [23], the group $J_{2}$ has an absolutely irreducible module of dimension 6 over $\mathbb{F}_{4}.$ Therefore, a split extension group having the form $4^{6}{:}J_{2}:= \overline{G}$ exists. In this paper, we consider this group. Our purpose is to determine its conjugacy classes and character table using the methods of the coset analysis together with Clifford–Fischer theory. We determine the inertia factors of $\overline{G}$ by analyzing the maximal subgroups of $J_{2}$ and the maximal of the maximal subgroups of $J_{2}$ together with other various information. It turns out that the character table of $\overline{G}$ is a $53 \times 53$ real-valued matrix, while Fischer matrices of the extension are all integer-valued matrices with sizes ranging from 1 to 8.