If you inserted the the authinfo code it means that is a REG to REG transfer and you can check it by yourself.
From the NIC site http://www.nic.it/web-whois/index.jsf insert the domain and the required control code, then check the "Status": if it's "pendingTransfer" means that the transfer is going correctly, if it's "ok" means that the transfer finished or could have failed, in that case, contact us.
The domain remains in the "pendingTransfer" for maximum 24 hourse. If within this time the Reg modify hasn't been explicitly cancelled from the new Reg or rejected from the old one, it is automatically approvved from the NIC and the transfer completed.
We can't accelerate this process: only you can do it, asking your actual provider to accept immediatly the transfer without waiting the 3 days (the provider isn't bound to do it).