PGL(3,C)
In this page, there is a list of all finite groups of $\rm{PGL}(3,\mathbb{C})$ acting on Project space $\mathbb{P}^2$,
(see section 10 in EQUIVARIANT BURNSIDE GROUPS AND REPRESENTATION THEORY)
$\textbf{Intransitive}$ subgroups: the representation is reducible and has the form $G=C_n \times G’$ where $G’ \subset GL_2$, which is cyclic, dihedral, $A_4$, $S_4$ or $A_5$.
$\textbf{Transitive but imprimitive}$ subgroups: the action is not transitive, but there is a nontrivial normal subgroup of $G$ acting intransitively. They are 4 types.
$\textbf{Primitive}$ subgroups of $\rm{PGL}(3,\mathbb{C})$ are isomorphic to one of the following groups:
- $A_5$ of order 60,
- $A_6$ of order 360,
- $PSL_2(\mathbb{F}_7)$ of order 168,
- the Hession group $ASL_2(\mathbb{F}_3)$ of order 216,
- A subgroup of Hessian group, $PSU_3(\mathbb{F}_2)$ of order 72,
- A subgroup of Hessian group, $C3:S3.C2$ of order 36.
The explicit generators of $\textbf{primitive}$ groups $G$ and the Burnside symbols of $G$ acting on $\mathbb{P}^2$ are given in the following pages: