command
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command / order
A command or instruction, usually from a person with authority. It is used almost exclusively in the phrase 'auf Geheiß von' (at the command of).
Auf Geheiß des Direktors wurde das Gebäude sofort evakuiert.
At the director's command, the building was evacuated immediately.
This word is used almost exclusively in the fixed phrase 'auf Geheiß von' + dative or 'auf jemandes Geheiß' (at someone's command). It expresses that an action happens because of an order.
Die Soldaten handelten auf Geheiß ihres Hauptmanns.
The soldiers acted at their captain's command.
'Geheiß' (command) is a very formal and somewhat outdated word. It is mainly found in written language, in historical contexts, or in formal reports, but hardly ever in everyday speech.
Auf Geheiß des Königs musste der Gefangene freigelassen werden.
At the king's command, the prisoner had to be released.
The word is related to the verb 'heißen', which in the past could also mean 'befehlen' (to command). The prefix 'Ge-' here points to the result of an action, that is, the command itself.
Er verließ den Raum auf ihr Geheiß.
He left the room at her command.
assignment / order / mission
'der Auftrag' and 'das Geheiß' can both mean that someone does something on instruction. 'der Auftrag' is the normal and neutral word: you receive a task and carry it out. 'das Geheiß' is much rarer and sounds elevated or formal. It almost always appears in the phrase 'auf Geheiß von' (at the command of). It then emphasizes that the instruction comes from an authority. Learners may confuse the words because both have to do with orders or instructions. In everyday life, however, people say 'Auftrag'; 'auf Geheiß von' fits more in formal reports or official texts.
Auf Geheiß der Leitung wurde der Termin kurzfristig verschoben.
At the management's command, the appointment was postponed at short notice.
Wir haben den Auftrag bekommen, die Räume bis Freitag zu prüfen.
We were given the assignment to inspect the rooms by Friday.