The finite irreducible monomial linear groups of degree \(4\) (Q1306897)

Let \(\text{GL}(n)=\text{GL}(n,\mathbb{C})\). The group of monomial matrices in \(\text{GL}(n)\) is denoted by \(M(n)\) and is isomorphic to the wreath product \(\mathbb{C}^\times\text{ wr }S_n\) where \(\mathbb{C}^\times\) denotes the multiplicative group of the complex field \(\mathbb{C}\). The base group of this wreath product is the group of non-singular matrices in \(\text{GL}(n)\) and is denoted by \(D(n)\). Therefore \(M(n)\) is isomorphic to the semi-direct product \(M(n)\cong D(n)\rtimes S_n\). In [The finite irreducible linear \(2\)-groups of degree \(4\), Mem. Am. Math. Soc. 613 (1997; Zbl 0927.20026)], the author lists all the finite irreducible 2-subgroups of \(\text{GL}(4)\) up to conjugacy. Since, in general, any finite \(p\)-subgroup of \(\text{GL}(n)\) is conjugate to a subgroup of \(M(n)\), this means that a listing of the finite irreducible 2-subgroups of \(M(4)\) is known up to \(\text{GL}(4)\)-conjugacy. In the present paper, the author extends his previous results to the whole of \(M(4)\), i.e., he lists all the finite irreducible subgroups of \(M(4)\) up to conjugacy in \(\text{GL}(4)\).