by order of someone—according to directions given by a proper authority:
- By order of the Minister a special operation of police is being carried out.
under the orders of someone—
1. (Military) commanded by a person; being smb.’s subordinate:
- At the battle of Waterloo he was under the orders of Marshal Grouchy.
2. = by order of someone:
- In some cases it may be possible to release the prisoner on probation, but this is permitted only under the orders of the Provincial Government.