Let $S(m,n)$ be the generalized symmetric group which is a wreath product of the cyclic group of order $m$, denoted here by $\mathbb{Z}_m$, and the symmetric group $S_n$. A standard unitary representation of $S(m,n)$ is given by the semi-direct product of the $n\times n$ permutation matrices and $n \times n$ diagonal matrices with $m$-th roots of unity entries. (These matrices are sometimes called generalized permutation matrices.)
For finite $m$ and $n$ all such unitary matrices form a finite subgroup of $U(n)$ of size $n! m^n$. I'll denote this group of unitary matrices as $M(m,n)$. One way to see that $M(m,n)$ is not a maximal finite subgroup of $U(n)$ is to note that $S(2m,n)$ is a finite group which has $S(m,n)$ as a subgroup. I'm interested in a slightly different question.
Let $U'\in U(n)$ be a unitary matrix which is not in $M(m,n)$ for any $m$ (in other words, $U'$ is not a generalized permutation matrix). Is the group generated by $M(m,n)$ (for some fixed $m$) and $U'$ always infinite?
Best Answer
I think the answer is "yes" when $m >6$. By arguments along the lines of Frobenius, Schur and Blichfeldt, if we set $G = \langle M(m,n), U^{\prime} \rangle $ and assume that $G$ is finite, then the non-scalar elements of $G$ whose eigenvalues all lies on an arc of length less than $\frac{\pi}{3}$ on the unit circle generate an Abelian normal subgroup of $G$, say $A$. When $m >6$, this Abelian group has rank $n$, and may be assumed to consist of diagonal matrices. In that case, $C_{G}(A)$ also consists of diagonal matrices (using that $A$ has rank $n$). By Clifford's theorem, we may conclude that the given representation of $A$ is now induced from a $1$-dimensional representation of a subgroup of $G$ of index $n$. In other words, $G$ now consists of monomial matrices.
I suppose, to be more precise, this argument shows that if $m > 6$, any finite unitary overgroup of $M(m,n)$ is conjugate (via a unitary matrix) to a finite group of monomial matrices ( what you call "generalized permutation matrices" are what I am calling monomial matrices).