Teaser 3047: Some permutations

From The Sunday Times, 14th February 2021 [link]

I gave Robbie three different, non-zero digits and asked him to add up all the different three-digit permutations he could make from them. As a check for him, I said that there should be three 3’s in his total. I then added two more [non-zero] digits to the [original three digits] to make [a set of] five digits, all being different, and asked Robbie’s mother to add up all the possible five-digit permutations of these digits. Again, as a check, I told her that the total should include five 6’s.

Given the above, the product of the five digits was as small as possible.

What, in ascending order, are the five digits?

I have changed the text of this puzzle slightly to make it clearer.