By 3 o'clock yesterday afternoon the dozen or so men and women who had been running the Inaugural Day TV network pool coverage out of a fourth-floor room on 13th Street had begun to relax.

"Close your mouth, kid," one of the technicians joked at Amy Carter on the monitor marked "White House Low Head On," which showed a closeup of a tired little girl on a cold reviewing stand where people were starting to pass around the blankets.

In another 45 minutes the parade would be over, and the cooperative effort that had sent coverage out to ABC, CBS, NBC, PBS, and TV networks in Great Britain and four other European countries would begin to shut down.

The room, located in a telephone company building, contained seven TV monitors - from which the director could order the pool picture of the moment "punched up" - and makeshift work tables at which most of the staff had been working since 7 a.m.

The pool system has become in a necessity a major TV events. There is simply not enough room for all the competing network cameras and equipment at critical sites such as the Inaugural stand, the reviewing stand and key parade route sites. So it is only the pool that gets to use a mobile unit that precedes the presidential party.

There were 22 different cameras available to the pool, and the images from seven of them were always simultaneously available to director Al Mifelow in the pool control room yesterday.

It was Mifelow who decided which single image would go out on the line as the pool picture - subject to veto by ABC's Daryll Griffin the producer. The pool picture has sound, but no commentary; that is added by later by reporters and anchormen.

It is producer Griffin, at the pool, leafing through a thick looseleaf notebook, who identifies for the networks the names of the march bands as they were picked up a long the route.

NBC yesterday crews manned the pool cameras at the Captiol, CBS crews were along the parade route, and ABC handled the six White House cameras.

In addition, any one of the three networks could also call for a specific shot from a pool camera, a request which Griffin almost instantaneously puts to a vote from the other two network coordinators operating from their own control boths.

NBC, at one time yesterday afternoon, wanted a close-up shot of the Capitol from the Monument and won, 2 to 1, despite the fact ABC had voted to continue a longer shote alreasy on the pool monitors.

"It's been very clean, everything's worked," said Ron Ogle, the coordinating producer, who was working his third Inaugural for ABC, the network which this year ran the network pool overall.

"The high point had to be the walk down Pennsylvania Avenue, of course," said Griffin "That was pretty good TV. The European networks apparently agreed. Most had decided to end their coverage with President Carter's address, but all over Europe last night the news shows opened with Amy Carter getting her shoes tied in the middle of Pennsylvania Avenue.

"Well, we did lose a cherrypicker," said Ogle. "A milk truck got the one in front of the White House very early this morning and wrecked a camera. Got it repaired though."

The pool control room was an engrossing keyhole to American history, but one that even the technical superiority of the networks couldn't permit the national audience to see. Just before 1 o'clock, the pool minitors showed President Ford heading for Air Force 26000 while a pool camera at the Capitol showed President Carter's waiting limousine with its ram-rod-stiff honor guard keeping vigil.

And out on the parade route at that same moment, other monitors showed that some ordinary citizens had begun to dance in the streets, and also that nobody was walking in the driveways of the White House. The city, like the nation, was in a graceful suspense.

There were also three monitors in the room that revealed what ABC, NBC and CBS were showing at the same moment. They were showing commercials.