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What would be a simple way, given a subgroup $G$ of $(\mathbf{Z}/m)^*$ (given as a list of integers coprime to $m$)

to define the corresponding (by Galois) subfield of the cyclotomic field $\mathbf{Q}(\zeta_m)$ ?

For example, $m=5$ and the subgroup is given by $L = [1, 4]$.