The Hydran Kingdom was the main government of the Hydrans. Its capital-world was Hydrax. It was ruled by a monarchy encompassing an unusually large royal family, comprising some 30,000 nobles, of which 1,200 are potential crown princes/princesses. Royal lines of succession weren't clear because royal families were polygamous. They were allied with the Mirak Star League because they had a common enemy: the Lyran Star Empire.
The Hydran nobility often got involved in the internal politics, but it really was the civil service who ran the kingdom.
Its most successful military corps was its fighter corps. Their ships were often known to carry, size for size, more fighters than their Federation or Klingon counterparts. They used squadrons of Stinger-class fighters to complement their ships' firepower. They also had a special force known as the Hydran Keepership. (TOS video game: Starfleet Command I & II)
As of 2388, the King of the Hydran Kingdom was S'Lentha VI. (RIS Bouteina: "New York Convention")
The Hydran Kingdom, contrary to the Interstellar Concordium, banned the emission of credit cards from its territory in an effort to regulate the financial sector. However, it was unclear if it accepted foreign credit cards or not. (RIS Bouteina: "Credit Card Scholarship")
Finally, the elite of the Hydran marines were known as Triads. (RIS Bouteina: "Hystelic Kingdom")
Hydran labor law treated paid test subjects as contractual jobs, and, as such, asked that credentials, work experience and background of job applicants are checked before test subjects can be hired. (Star Trek: False Vacuum: "Wagon Train to the Stars")