A group of prominent Fairfax County residents has filed a $1.1 million lawsuit against a national cable television company, charging they were cheated out of the opportunity to compete for the county's lucrative cable TV franchise.

In a suit filed in U. S. District Court in Alexandria, the group's six members charge that officials of Cross Country Cable of Suffolk, Va., backed out of their pledge to seek the county's cable franchise, thus depriving local investors of cash payments that they had been promised by company officials. The suit seeks $100,000 to cover alleged financial damages to the local investors and $1 million in punitive damages.

"We felt we had an agreement, so to speak," said Glenn W. Saunders Jr., former Fairfax city manager and one of the plaintiffs. "They decided unilaterally not to go for the franchise and forgot about all the discussions we had."

Officials at Cross Country Cable, which dropped out of the Fairfax cable competition early this year, could not be reached for comment.

According to the suit, Cross Country Cable in 1980 recruited the group, which includes former county supervisors Harriet F. Bradley and William S. Hoofnagle, to act as its local franchising advisers. The plaintiffs allege that each member of the group was promised stock amounting to one percent of the company's Fairfax operation, and a commensurate amount of the local profits if they were successful in winning the franchise.

After widespread criticism of cable lobbying by local citizens, the complaint says, that arrangement was changed to one pledging that each of the local advisers would be paid $5,600 annually for their services regardless of whether Cross Country got the franchise.

Subsequently, according to the complaint, Cross Country Cable abandoned its plans to apply for the county's franchise and agreed to help construct or operate a cable system for competing Fairfax Telecommunications Inc., a subsidiary of Denver-based Tele-Communications Inc. At the same time, Cross Country allegedly paid its local participants $1,500 apiece for their efforts on Cross Country's behalf.

That decision, the local group alleges, "deprived these plaintiffs of their right to participate in whatever benefits will accrue to Cross Country officials from their agreement" with FTCI or TCI.