A two$-$dimensional crystal has plane group $p3m1$ with a lattice constant $a=6.$ It has two types of atoms, A and B$,$ both with diameter $1\mathrm{.}5.$ Wyckoff positions are occupied as follows:

A in site $1b,B$ in site $1c.$

$($a$)$ Make a scale drawing of the structure, covering an area of at least four conventional unit cells.

$($b$)$ What are the coordination numbers of the A toms and of the $B$ atoms.

$($c$)$ What is the center$-$to$-$center distance between A and B atoms?

$($d$)$ Which of the five plane lattices does this structure possess?

$($e$)$ If Wyckoff sites $1b$ and $1c$ were occupied by the same species, and these were the only sites occupied, would the plane group of the resulting structure still be $p3m1?$ Explain your answer.