The Transitivity Restriction (6 Groups for 83)

Because 83 is prime, the number of options for transitive permutation groups is very limited. Since the underlying group for the eyes is most likely transitive, based on the chaining graphs for the isomorphs covering nearly all ciphertext symbols, this restricts the options for possible groups down to only 6 options. These options are $C_{83}$ (cyclic), $D_{166}$ (dihedral), $C_{83}:C_{41}$ (subgroup of AGL), $C_{83}:C_{82}$ (AGL), $A_{83}$ (alternating), and $S_{83}$ (symmetric). The cyclic option is directly ruled out because it is commutative, and we see chaining conflicts indicating non-commutativity. The dihedral option was directly ruled out based on conclusions about possible element orders using the main isomorphs in the first three messages. The AGL options have not been completely ruled out, but are not generally able to produce shared sections after a differing first character, unless the initial states of all the messages are fine-tuned to allow for an immediate resync. So what remains are the alternating and symmetric groups consisting of general permutations, rather than one of the smaller groups that would have made this much easier to solve.