The designation of George P. Shultz as secretary of State strengthens the already excellent Reagan administration connections of Bechtel Group Inc., one of the world's biggest construction companies, a family-owned California firm with close ties to the Republican political establishment, the Saudi Arabian government, organized labor and the nuclear power industry.

Shultz, who was vice chairman of Bechtel's three operating companies for six years and who became president in 1980, was preceded into the administration by two other senior Bechtel executives, Caspar W. Weinberger, now secretary of Defense, and W. Kenneth Davis, deputy secretary of Energy.

Bechtel, based in San Francisco, is a far-flung construction empire controlled by a small group of little-known men. The company, which specializes in nuclear power plants, nuclear waste disposal, and large-scale construction projects in remote parts of the world, reported billings of $11.4 billion in 1981.

According to Paul W. Cane, vice president for public affairs, Bechtel was taken by surprise by Shultz's appointment and has named no successor. "We just learned about this today," he said.

He said Bechtel always has understood that Shultz, a former secretary of Labor and secretary of the Treasury, "would respond unhesitatingly if he were called once more to serve his country. We are confident that his enormous talent will be a great asset to President Reagan and the nation."

In many ways, the job that Shultz leaves behind is similar to the one he is taking up, requiring political acumen and the ability to deal with a vast array of political officials, foreign governments and international subcontractors. The position he is leaving is not an engineering post. Bechtel's chief engineer is Stephen Bechtel Jr., grandson of the founder and chairman of the company.

Because Bechtel is a privately owned company that sells no stock to the public, it does not release profit figures. Analysts who have studied the company say there is no doubt that it has been steadily and spectacularly successful since it was founded by railroad construction worker Warren A. Bechtel in 1898.

Bechtel has about 120,000 employes, and is controlled by members of the Bechtel family and by about 60 senior executives. Shultz, as one of those executives, was drawing a salary of almost $500,000 a year and also receiving stock, according to records filed with the State of California.

The Bechtel family also owns Sequoia Investments Inc., a private investment company, which holds a majority interest in the brokerage house of Dillon, Read & Co. Shultz, who was chairman of Sequoia, was also a member of the board of Dillon, Read. He presumably will have to resign from both positions to return to the Cabinet. Nicholas Brady, managing director of Dillon, Read, recently was appointed to the U.S. Senate to fill the unexpired term of Sen. Harrison Williams (D-N.J.).

The list of projects built or managed by Bechtel includes some of the most famous and most complex construction jobs ever undertaken: Hoover and Bonneville dams, the San Francisco-Oakland Bay Bridge, the Transarabian pipeline, the government and foreign ministry buildings in Brunei on the north coast of Borneo, the industrial city of Jubail, Saudi Arabia, and dozens of hydroelectric and transportation developments. Bechtel is the supervising contractor on the Washington Metro subway system.

Bechtel also has built nuclear power plants in Pennsylvania, Mississippi, Michigan, Oregon, Florida, Washington and other states, and is heavily involved in several nuclear-waste disposal projects.

In a recent report to its employes, Bechtel noted that "the nuclear power business is seeing difficult times with little prospect of quick turnaround," but added that "the balance of the nuclear fuel cycle represents an area of possible growth and technological advancement." This advancement is "heavily dependent on the support" of the Reagan administration, it said.

Bechtel's long association with Saudi Arabia and other Arab countries often has made the company the target of critical scrutiny by supporters of Israel. In 1976 the Justice Department accused Bechtel of illegally cooperating with the Arab nations' blacklist of corporations that do business with Israel.

Bechtel settled the complaint in 1977 by accepting a consent decree in which it agreed not to violate antiboycott laws. Bechtel later claimed that the government had altered the decree and asked to be excused from complying with it, but the Supreme Court let stand last December a ruling that the decree was valid and must be followed.

In his annual report to employes, Stephen Bechtel Jr. predicted revenue growth of 18 percent for 1982. He said earnings rose last year, even when adjusted for inflation, and predicted that they will continue to go up.

He said Bechtel will benefit from a resurgent economy, and noted that "the U.S. government's avowed support for the electric utility industry, and a growing public awareness of its financial and regulatory difficulties, can be expected to add to the upward movement of the domestic economy."