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The political followers of Thomas Jefferson took up the name "Democratic-Republicans," preferring this over the previous term "Anti-Federalists" since the Constitution had already be drafted and ratified.
The Constitution does not directly mention secession, but the Supreme Court has consistently interpreted the Constitution to be an indestructible union. There is no legal basis a state can point to for unilaterally seceding. So basically, no, a state can not secede, it is blatantly illegal.