when the one who gives the permission is mentioned 

when the one who’s giving the permission is unknown or isn’t mentioned 

in i, you, we and they, let, don’t let and don’t bla let is used but in she, he and it lets, doesn't let and doesn't bla let is used 

when is i, you, we or they, am and are are used but in she, he and it is is used

“he lets her go to the dance class” “she is allowed to go to the dance class”

how would you say this sentence en both ways “he gived her permission to go to dance class”

Let and allowed to