On a group of the form $2^{11}:M_{24}$

The Conway group $Co_{1}$ is one of the $26$ sporadic simple groups. It is the largest of the three Conway groups with order $4157776806543360000=2^{21}.3^9.5^4.7^2.11.13.23$ and has $22$ conjugacy classes of maximal subgroups. In this paper, we discuss a group of the form $\overline{G}=N\colon G$, where $N=2^{11}$ and $G=M_{24}$. This group $\overline{G}=N\colon G=2^{11}\colon M_{24}$ is a split extension of an elementary abelian group $N=2^{11}$ by a Mathieu group $G=M_{24}$. Using the computed Fischer matrices for each class representative $g$ of $G$ and ordinary character tables of the inertia factor groups of $G$, we obtain the full character table of $\overline{G}$. The complete fusion of $\overline{G}$ into its mother group $Co_1$ is also determined using the permutation character of $Co_1$.