Grammar Tables: 'let,allow,permit
Both 'let' and 'allow' can be used to talk about giving and refusing permission.

Let

subject let object infinitive or prepositional phrase
They let me go out and get a drink
They wouldn't let me back in
Let me make a note of the phone number.

Allow

subject allow object infinitive with 'to'
They allowed me to go out and get a drink
She won't allow anyone to smoke in the office
Allow (passive)
passive - subject +verb 'to be' allowed infinitive with 'to'
He wasn't allowed to go out and get a drink
Nobody is allowed to smoke in the office

Permit

Smoking is not permitted in this room.
You should not permit the use of the computer.