Siblings for a Galois group (reviewed)

An abstract group $G$ may be a transitive subgroup of $S_n$ for different $n$ and even in different (non-conjugate) ways for a given $n$. Each action is identified by a label of the form nTt where $n$ is the degree and t is the T-number classifying the action.

In terms of the Galois correspondence, the group $G$ corresponds to a degree $n$ extension $F/K$ where $F=K(\alpha)$, and $G$ is the Galois group of the splitting field of the monic irreducible polynomial for $\alpha$. The siblings correspond to sibling fields, which are not isomorphic to $F$, yet have the same normal closure.

There can be more than one action with the same transitive classification. This corresponds to non-isomorphic fields with the same degree, Galois group, and Galois closure. We indicate multiplicity using the notation "nTt x k" where there are k non-conjugate subgroups such that the action is nTt.