Find regular respresentation of quaternion group $Q_{8}$. 
Find regular respresentation of quaternion group $Q_{8}$.

I am not very familiar with regular respresentations. As far as I know, I need to find all multiplications of elements of group $Q_{8}$ and than observe those multiplications and conclude what element of group $S_{n}$? Is that true? I am beginner in this scope so any hint helps! Thanks.
 A: By definition of the regular representation, the degree of the representation is the group order i.e. $8$. If you want to explicitely find the matrices (over any field $k$), consider the action of the element on each of the other elements in the group. 
Concisely, this means (as reuns noted in the comment) 
$$\varrho(g)_{ij} = \begin{cases}
1 & g.g_j = g_i \\
0 & ~ \text{otherwise}
\end{cases}
$$
(Again, as reuns noted), it suffices to explicitely compute the $8 \times 8$ matrices for the two generators $i,j$ where $Q_8 := \langle i,j \vert i^4 = 1, i^2 = j^2, jij^{-1} = i^{-1} \rangle$. Then the representing matrices for the other elements of the group can be computed via matrix multiplication.

If you are learning representation theory of finite groups and want some explicit examples, have a look at James & Liebeck. 

Also, if you are aquainted with representation theory via $\mathbb{F}G$-modules, remember that the regular representation is defined as the $\mathbb{F}$-vector space with basis $g_1, \ldots, g_n$ where $\vert G \vert = n$. Then it becomes clear why the coefficients of the representing matrix are determined that way.
