For my research, I need to compute some field of invariants for a specific subgroup of $GL_2(F)$.
My field $F$ has characteristic different from $2$ and contains a primitive $4th$-root of $1$, denoted by $i$.
If necessary, I am ready to assume stronger assumptions on $F$ (even if I would like to avoid it)
The group I am interested is the following. Let $A=\pmatrix{i& 0\cr 0 & 1},B=\pmatrix{1 & 0 \cr 0 & i}, C=\pmatrix{0 & 1\cr 1 & 0}.$
Set $G=\langle A,B,C \rangle$. I am interested in computing $F(V)^G$ in the two following cases:
$V=M_2(F)$ and $G$ acts by conjugation
$V=M_2(F)\times M_2(F)$ and $G$ acts by simultaneous conjugation.
More precisely, I would like to have a good description of the structure of $F(V)^G/F$: is is purely transcendental or not? Can I obtain a transcendence basis? If it is not purely transcendental, can we describe it explicitely as a finite extension of a purely transcendental subextension? etc...
I am totally new to the theory of invariants. I looked at some books and was able to compute some invariants in case 1):
$F(V)^{\langle A,B\rangle}=F(a,d,b^4,bc)$ and $F(V)^{\langle C\rangle}=F(a+d,b+c, (a-d)(b-c), bc)$.
However, I am unable to compute $F(V)^G$. Note that $G=\langle A,B \rangle\rtimes \langle C\rangle\ \simeq (\mu_4\times\mu_4)\rtimes\mu_2,$ where $-1\in\mu_2$ acts by swapping the coordinates.
I would be happy if someone could help me with these computation, at least in case 1., or suggest some useful techniques to achieve it.