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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)$$f^ \sigma (x):=f(\sigma^{-1}x)$, 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 $x=(x_1,...,x_n)$.

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 $\mathbb C[x,y,z,]^H=\mathbb C[f_0,f_1,f_2]$ ?

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 $\mathbb C[x,y,z,]^H=\mathbb C[f_0,f_1,f_2]$ ?

For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x_1,...,x_n)$.

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 $\mathbb C[x,y,z,]^H=\mathbb C[f_0,f_1,f_2]$ ?

reduced huge formula and fixed ill-placed period
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For $\sigma \in GL_n(\mathbb C)$$\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

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

For a subgroup $G$ of $GL_n(\mathbb C)$$\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 $f_0(x,y,z)=x^3+y^3+z^3+6xyz, f_1(x,y,z)=3(x^2y+y^2z+z^2x), f_2(x,y,z)=3(xy^2+yz^2+zx^2)$.$f_0(x,y,z)=x^3+y^3+z^3+6xyz$, $$f_1(x,y,z)=3(x^2y+y^2z+z^2x),\quad f_2(x,y,z)=3(xy^2+yz^2+zx^2).$$

Let $G=\{\sigma \in GL_3(\mathbb C) : f_0^\sigma =f_0, (f_1^\sigma)^3=f_1^3, (f_2^\sigma)^3=(f_2)^3 \}$$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 GL_3(\mathbb C) : f_0^\sigma =f_0, f_1^\sigma=f_1, f_2^\sigma=f_2 \}$$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 $\mathbb C[x,y,z,]^H=\mathbb C[f_0,f_1,f_2]$ ?

For $\sigma \in GL_n(\mathbb C)$ 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

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

For a subgroup $G$ of $GL_n(\mathbb C)$, 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 $f_0(x,y,z)=x^3+y^3+z^3+6xyz, f_1(x,y,z)=3(x^2y+y^2z+z^2x), f_2(x,y,z)=3(xy^2+yz^2+zx^2)$.

Let $G=\{\sigma \in GL_3(\mathbb C) : f_0^\sigma =f_0, (f_1^\sigma)^3=f_1^3, (f_2^\sigma)^3=(f_2)^3 \}$ .

Let $H=\{\sigma \in GL_3(\mathbb C) : f_0^\sigma =f_0, f_1^\sigma=f_1, f_2^\sigma=f_2 \}$.

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 $\mathbb C[x,y,z,]^H=\mathbb C[f_0,f_1,f_2]$ ?

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 $\mathbb C[x,y,z,]^H=\mathbb C[f_0,f_1,f_2]$ ?

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Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$

For $\sigma \in GL_n(\mathbb C)$ 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

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

For a subgroup $G$ of $GL_n(\mathbb C)$, 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 $f_0(x,y,z)=x^3+y^3+z^3+6xyz, f_1(x,y,z)=3(x^2y+y^2z+z^2x), f_2(x,y,z)=3(xy^2+yz^2+zx^2)$.

Let $G=\{\sigma \in GL_3(\mathbb C) : f_0^\sigma =f_0, (f_1^\sigma)^3=f_1^3, (f_2^\sigma)^3=(f_2)^3 \}$ .

Let $H=\{\sigma \in GL_3(\mathbb C) : f_0^\sigma =f_0, f_1^\sigma=f_1, f_2^\sigma=f_2 \}$.

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 $\mathbb C[x,y,z,]^H=\mathbb C[f_0,f_1,f_2]$ ?