Civil magistrates; government functionaries. The word does not appear in Johnson's 1755 Dictionary, but he does include the cognate verb, to syndicate:
To judge; to pass judgment on, to censure. An unusual word.
In the Genevan political system, according to the account in the 4th edition of the Encyclopaedia Brittanica (1797), there were only four syndics chosen from among the magistrates and these held the greatest authority in the republic. Geneva's syndics passed a sentence of exile on Jean-Jacques Rousseau in 1762, a time we might suppose included in Alphonse Frankenstein's "many years." Mary Shelley mentions the event in History of a Six Weeks' Tour, Letter II.