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 or monomial matrices over the field $\mathbb{C}$.)
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 $M(m,n)$.
Recently, in Are generalized symmetric groups maximal finite groups (in a certain sense)? Geoff Robinson showed for any fixed $n$ and $m\ge 7$ the group $M(m,n)$ and any additional unitary (in $U(n)$) which is not a generalized permutation generate an infinite group.
This post is to tie up some loose (at least to me) ends. I want to know if the following groups are maximally finite (in this sense) at a smaller $m$.
(1) Let $U_4 \in U(4)$ be a unitary matrix which is not in $M(m,4)$ for any $m$ (in other words, $U_4$ is not a generalized permutation matrix). Is the group generated by $M(4,4)$ and $U_4$ always infinite?
(2) Let $U_8 \in U(8)$ be a unitary matrix which is not in $M(m,8)$ for any $m$. Is the group generated by $M(2,8)$ (a signed permutation group) and $U_8$ always infinite?
