We know that every $2\times 2$ matrix in $PGL(2, Z)$ of order 3 is conjugate to the matrix [ \left (\begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) ]. I am interested in finding out to what extent this holds for $3\times 3$ integer invertible matrices.

In other words how many conjugacy classes of order 3 matrices in $PGL(3, \mathbb Z)$ are there?

We know that every $2\times 2$ matrix in $PGL(2, \mathbb{Z})$ of order $3$ is conjugate to the matrix $$ \left( \begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) $$. 

I am interested in finding out to what extent this holds for $3\times 3$ integer invertible matrices.

In other words how many conjugacy classes of order 3 matrices in $PGL(3, \mathbb{Z})$ are there?

We know that every $2\times 2$ matrix of order 3 is conjugate to the matrix [ \left (\begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) ]. I am interested in finding out to what extent this holds for $3\times 3$ integer invertible matrices.

In other words how many conjugacy classes of order 3 matrices in $GL(3, \mathbb Z)$ are there?

We know that every $2\times 2$ matrix in $PGL(2, Z)$ of order 3 is conjugate to the matrix [ \left (\begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) ]. I am interested in finding out to what extent this holds for $3\times 3$ integer invertible matrices.

In other words how many conjugacy classes of order 3 matrices in $PGL(3, \mathbb Z)$ are there?