D.C. City Council Chairman Arrington Dixon found a convenient way to get to inaugural events Monday night through the traffic jams that clogged city streets, impeding the thousands of other celebrants trying to get to the same festivities.

Dixon simply summoned police for a little chauffeur service.

Some 1st District police officers, whose beats include the District Building and inaugural parade area of Pennsylvania Avenue, were still grumbling yesterday about the severl hours of private service, complaining that it came at a time when the force was stretched thin trying to keep order in a city overflowing with visitors and celebrating residents.

In addition to massive traffic problems, police documents from the 1st District show that the downtown area had 54 "serious crimes" reported Monday night and Tuesday, nearly twice the number recorded in any of the other six districts. The crimes included 19 thefts from autos, 9 roberies, 12 larcenies and 6 stolen autos.

The officer Monday night had already worked a full day and then he had to spend several hours taking him [Dixon] from party to party until after midnight," one police officer complained. Dixon said yesterday that he does not remember how many events he attended Monday night, but acknowledged use of the police driver and the vehicle that otherwise could have been on patrol.

The officers also criticized Dixon for using another police car on inauguration day, although a police department official said Dixon's trip from the District Building to Capitol Hill was "entirely proper." But the official said he did not know about the use of a police car Monday night. "I might question that myself," the official said.

The council chairman said yesterday that his own city car -- a recycled police cruiser rejected for use by the police department -- was not running properly and that using the police car and driver was "consistent with" a briefing on security given him by police officials.

At the briefing, Dixon was told inauguration day security would be extraordinarily tight and that the city was limiting automobile passes to about 500 in an area around the Pennsylvanai Avenue parade route.

The passes were sharply reduced from President Carter's inauguration in 1977, when 1,700 passes were issued. Police officials said security was tightened because of several threats on Reagan's life, and the tensions growing out of the expected release of the American hostages of Iran.

"That car Dixon used could have been in service," a police offier said. "He could have gotten to his own parties."