I want to list the elements of
So from my understanding, with
But I'm having trouble trying to make senses of the two other conditions, and how to properly translate them into the table.
Notice first that .
From the relation it follows that . Using that, in any product of a's and b's you can always push the a's to the left of the b's. For example . In that way, you can express any element of the group as , with and . That gives you 24 elements of the group, and you still have a fair amount of work to do if you want to write down the whole multiplication table.