The phrase "of Satan" does not mean that the assembly was called for the overt purpose of worshipping Satan, but that despite it being Jewish in name, its people were actually serving Satan. This is readily evident from both the use of the term itself and the reference to Jews who were not truly Jews associated with it. This usage is neither cryptic nor arcane. Jesus accused Jews who were seeking to harm him of being the children of Satan, so the connection between opposition to Christ and the Devil was not new.
As I understand it, these were Jews who were not willing to seriously vet his claims of Messiahship--claims that were primarily substantiated by incredible, miraculous feats, and ultimately, rising from the dead. They could not get past his illegitimacy (his earthly father was not his actual father), nor their place of assumed privilege with God that came by virtue of being born Jewish (Abraham's descendants). They had so invested themselves in an approach to Judaism that they were not willing to acknowledge the Jewish Messiah because of that investment. Their minds were made up to reject and resist him (even to kill him) before his case could even be argued.
Callous unbelief would be a fitting description of their attitude toward Jesus. From that position, and with whatever influence they had with those who had political power, they relentlessly pursued a course of rejection, obfuscation, and calumny against Christ (and later, against his followers). This sort had, at the time of the writing of the Apocalypse, famously done so in Jerusalem and in Rome (Suetonius 25.4), as well as in Pisidia, Lycaonia and Thessalonica. This is the sort that Christ was referring to by the moniker, "Synagogue of Satan," when he warned the churches in Smyrna and Philadelphia through the Apocalyptic letters.
Now, the Westminster Confession (25:5) attributes to that moniker the condition of lacking proper discipline in a Christian congregation which allows heresies to spread unchecked. Such a church, in effect, becomes a "synagogue of Satan", but that is not even close contextually to how the term was used scripturally. Unfortunately, that is not the worst hatchet job on the phrase out there--there is a lunatic fringe which attaches a labyrinthine conspiracy to attain world domination on the part of some very selective, secretive, and fabulously wealthy Jews to the term. Nuts, truly, are not just a garnish for salads!