|Permit, allow and let are verbs which mean to give consent to. Permit however is the most formal among these three verbs. When there's an object, both permit and allow can be followed by the object and then an infinitive. Look at these examples: My mother permitted/allowed me to go to the party; Anna was not permitted/ allowed to go out last night. Without an object, permit and allow are followed by a gerund as in the following sentences: Our company does not permit/ allow smoking in the building. Her school permits/allows eating in the classroom during break time.|
Permit and allow can can also be used in a passive sentence form as in these sentences: Carrying firearms in public is not permitted/ allowed; Bringing packed lunch is permitted/allowed by the school.
While let has the same meaning as permit and allow, it is usually used differently. Let is the least formal of the three. In the sentence: Please let the cat out, the infinitive doesn't need 'to'. Compare these examples: I will permit/ allow you to join your friends tonight; I will let you join your friends tonight. Unlike permit and allow, let cannot be used in a passive sentence.