An action at law may be maintained in the name of the state, upon the information of the district attorney, or upon the relation of a private party against the person offending, in the following cases:
(1) When any person usurps, intrudes into, or unlawfully holds or exercises any public office, civil or military, or any franchise within this state, or any office in a corporation either public or private, created or formed by or under the authority of this state; or,
(2) When any public officer, civil or military, does or suffers an act which, by the provisions of law, makes a forfeiture of the office of the public officer; or,
(3) When any association or number of persons acts within this state, as a corporation, without being duly incorporated.