The output of the nation's factories mines and utilities grew by a modest 0.2 per cent in December as the coal strike halved industrial production last month, the Federal Reserve Board reported yesterday.

Economists said that the modest December rise reflects the slower production of goods in the final three months of the year as businesses decided to use up some of their inventories.

The sloer growth in industrial production in the last quarter - a revised 0.4 per cent in November and 0.2 per cent in October - was accompanied by large gains in employment and a drop in the unemployment rate to a three-year low of 6.4 per cent in December.

The large gains in employment reflect growing confidence in the economic recovery by businesses which now seem more willing to add employees than have existing employees work overtime.

Government analysts have said that, although final sales of goods will be very high in the last quarter of 1977, overall economic growth (as measured by the gross national product) will be slower than in the third quarter because many sales came from stocks that were not replenished in the last quarter.

However, production should pick up early this year as businesses find themselves required to add to their inventories.

The Federal Reserve said that moderate production increases were recorded in nearly all areas, including "home goods, consumer nondurable goods, business equipment, construction supplies, and durable and non durabe materials.

The nation's central bank reported that output of consumer goods rose by 0.3 per cent last month, "reflecting moderate increases in home goods (such as appliances and carpeting) and nondurable goods such as food, staples and clothing."

The Fed said that, because of weaker-than expected to build cars at about the same pace as in December, the agency said.

In part because of settlement of a strike in the aerospace industry, production of business equipment rose by 0.4 per cent last month, a gain over the previous two months.

Output of intermediate products, including construction supplies, rose by 0.9 per cent, indicating that building activity continued at a strong pace in December.

The coal strike, which began in early December, sharply reduced production of the fuel last month and triggered an overall 3 per cent decline in production of energy materials, the Federal Reserve reported.

November industrial production was buouyed in part by utilities and other users stockpiling coal in anticipation of the Dec. 6 walkout, which has affected about half the nation's coal production.

Because of the coal decline, overall production of materials fell 0.1 per cent.

Economists generally watch factory production because changes there usually translate quiclky into increased jobs. In recent months, relatively small increases in production appear to have resulted in substantial hiring gains, although the unemployment surveys measure more than the goods-producing industries gauged by the industrial production index.

The agency said that industrial production stood at 139.6 per cent of its 1967 average last month, which is 5 per cent higher than a year ago.