$\begin{matrix} & e& a& b& c& d&f \\ e& e& a & b& c& d&f \\ a& a& b& e& d& f&c \\ b& b& e& a& f& c&d \\ c& c& f& d& e& b&a \\ d& d& c& f& a& e&b \\ f& f& d& c& b& a&e \end{matrix}$
This is the group table I have created for $S_3$ group. Now, I want to find out the regular representation for the group element $c$ or what is $D_{ij}$.I also have to write every steps in detail.
Come on, this is really not difficult. The left regular representation of $c$ is the matrix of the left translation by $c$ in the algebra $\mathbb{Z}S_3$. Thus, according to your table:
$ca=f$, $cb=d$, $cc=e$, $cd=b$, $ce=c$, $cf=a$.
In other words,
$ca=0a + 0b + 0c + 0d + 0e + 1f$, $cb=0a + 0b + 0c + 1d + 0e + 0e$, $cc=0a + 0b + 0c + 0d + 1e + 0f$, $cd=0a + 1b + 0c + 0d + 0e + 0f$, $ce=0a + 0b + 1c + 0d + 0e + 0f$, $cf=1a + 0b + 0c + 0d + 0e + 0f$,
so the matrix is
$\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}$.