The most common and most likely answer to this crossword clue is the 3 plus 4 letter word THE BEST
I've seen this crossword clue in Financial Times
T
H
E
B
E
S
T
Buddy explains!
The straight part
We know that "First-class/punishment once given in sixes, apparently" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case first-class/punishment once given in sixes, apparently = the best.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.