Cyclic Groups

When you patch several multiplication tables of the same cyclic group, you see a diagonal band pattern. For instance, for the additive group Z/6Z, you get

1. Can you use any order of the elements in the header to obtain this band pattern? If not, specify the condition on the order of the elements.
2. Is it true that any group that displays this pattern, must be a cyclic group? If so, why?

You can use the table below to experiment. You can enter for instance cyclic groups Z/nZ (use notation Cn) or non-cyclic groups like the group S₃ (notation S3) or A₄ (notation A4).