What is a system of government by groups or officials?

An "oligarchy" is a system of government by groups or officials. It is a government that is run be several people as a group. It is distinguished from a "monarch", which is a government run by one person.
A system of government run by groups or officials is bureaucratic.
A bureaucracy is a system of government by groups of officials. An autocracy is a system of government where one person rules with total power.