How fill in this multiplication table? The following multiplication table was given to me as a class exercise. I should have all the necessary information to fill it completely in. However, I'm not sure how to take advantage of the relations I am given to fill it in?
The Question
A group has four elements $a,b,c$ and $d$, subject to the rules $ca = a$ and $d^2 = a$. Fill in the entire multiplication table. 
\begin{array}{c|cccc}
\cdot & a & b & c & d \\ \hline
 a&  &  &  & \\ 
 b&  &  &  & \\ 
 c& a &  &  & \\ 
 d&  &  &  & a
\end{array}
I imagine I might proceed like this: 
To find $ab$, write $a = d^2$ and thus $ab = d^2b = db\cdot b$....but my chain of reasoning always stops around here. 

Since each element has an inverse, each row and each column of the table must contain all four elements. 
After filling in the row and the column of the identity element $c$, we have three $a$'s in the table, and it follows that the only place for the last $a$ is given by $b^2=a$. Trying to distribute four $d$'s in the table, we are led to $ab=ba=d$. The only place for the remaining $b$'s is now given by $ad=da=b$, and the remaining products are then $=c$.
