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If we look at the Dihedral 4 group. There exists a trivial matrix representation that also makes it very easy to define group action on vectors: simply use 2x2 rotation matrices for the cyclic part and a reflection matrix for the reflection part. Then group action will be given by simply multiplying representation matrix right onto coordinate vector.


However I would like to build a matrix representation using a simpler field. Say for example $\mathbb Z_2$.

Now what I know is that the "cyclic part" of this group can be done with permutation matrices, generator:

$$C_G = \left[\begin{array}{cccc}&1&&\\&&1&\\&&&1\\1&&&\end{array}\right]$$

The non-reflected rotations could then have matrices $\{{C_G}^0, {C_G}^1,{C_G}^2,{C_G}^3\}$

But how can we do the reflection part? And further: How can we afterwards define group action (working on $\mathbb R^2$) with this new matrix representation?

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Generally, if $G$ embeds into $S_n$ then for any field $F$, $G$ embeds into $GL_N(F)$ as follows (note that I prefer right actions, but everything can be done on the left):

For convenience, identify $G$ with some isomorphic subgroup of $S_n$ so $G\le S_n$.

Fix a basis $e_1,\ldots,e_n$ of $F^n$. An embedding $\psi:G\hookrightarrow GL_n(F)$ is then given by $e_i\psi(g)=e_{i^g}$ (extended linearly).

In your case, a choice for one reflection would be

$$\left(\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{matrix}\right)$$

The other reflections would be the conjugates of this by your rotations.

An action through linear maps on $\mathbb{R}^2$ is equivalent to an isomorphism with a subgroup of $GL_2(\mathbb{R})$ which you noted at the start of your question.

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