For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x_1,...,x_n):=f(x_1^\sigma ,...,x_n^\sigma)$, where

$$\begin{pmatrix} x_1^\sigma \\ x_2^\sigma \\ \vdots \\ x_n^\sigma \end{pmatrix}= \sigma^{-1} \begin{pmatrix} x_1 \\ x_2 \\\vdots \\ x_n \end{pmatrix}.$$

For a subgroup $G$ of $\mathrm{GL}_n(\mathbb C)$, let $\mathbb C[x_1,...,x_n]^G :=\{f\in \mathbb C[x_1,...,x_n] : f^\sigma =f ,\forall \sigma \in G\}$.

Now let $f_0,f_1,f_2 \in \mathbb C[x,y,z]$ be defined as $f_0(x,y,z)=x^3+y^3+z^3+6xyz$, $$f_1(x,y,z)=3(x^2y+y^2z+z^2x),\quad f_2(x,y,z)=3(xy^2+yz^2+zx^2).$$

Let $G=\{\sigma \in \mathrm{GL}_3(\mathbb C) : f_0^\sigma =f_0,\; (f_1^\sigma)^3=f_1^3,\; (f_2^\sigma)^3=(f_2)^3 \}$ .

Let $H=\{\sigma \in \mathrm{GL}_3(\mathbb C) : f_0^\sigma =f_0,\; f_1^\sigma=f_1,\; f_2^\sigma=f_2 \}$.

What are the orders of $G$ and $H$ ? Is $H$ a proper subgroup of $G$ ? How to show that every matrix in $G$ is unitary ? Is it true that $\mathbb C[x,y,z]^G=\mathbb C[f_0,f_1^3,f_2^3]$ ? Is it true that $\mathbb C[x,y,z,]^H=\mathbb C[f_0,f_1,f_2]$ ?