I guess one answer is there's an isomorphism between your group and

$$SL_n \mathbb C \times_{\mathbb Z_nZ /n\mathbb Z} SO_2$$

My notation means take the product and mod out by the diagonally embedded copy of $\mathbb Z_n$. Z/n \mathbb{Z}$. The embedding of$\mathbb Z_n$Z/n\mathbb Z$ in $SO_2$ is as the cyclic subgroup of order $n$, and the embedding in $SL_n \mathbb C$ is the matrices of the form $\lambda I$ where $\lambda \in S^1$ is an $n$-th root of unity and $I$ is the identity matrix in $SL_n \mathbb C$.

So it's almost a direct product of two fairly nice groups.

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I guess one answer is there's an isomorphism between your group and

$$SL_n \mathbb C \times_{\mathbb Z_n} SO_2$$

My notation means take the product and mod out by the diagonally embedded copy of $\mathbb Z_n$.

The embedding of $\mathbb Z_n$ in $SO_2$ is as the cyclic subgroup of order $n$, and the embedding in $SL_n \mathbb C$ is the matrices of the form $\lambda I$ where $\lambda \in S^1$ is an $n$-th root of unity and $I$ is the identity matrix in $SL_n \mathbb C$.

So it's almost a direct product of two fairly nice groups.