the six diffeomorphic compact orientable classes are given in Table \[Tb:OESF\], where an isometry in Euclidean 3-space is denoted by $(A,a)$, $a$ is a vector and $A$ is an orthogonal transformation, and the action is given by $$\label{action}
(A,a) : p \mapsto Ap + a \; ,$$ for any point $p$. The orientation preserving orthogonal transformations that appear in the classification of the Euclidean space forms take the matrix forms $$\begin{aligned}
\label{Rot3}
A_1 = \left( \begin{array}{ccc}
               1 &  0 & 0 \\
               0 & -1 & 0 \\
               0 &  0 & -1
      \end{array} \right) \; , & 
A_2 = \left( \begin{array}{ccc}
              -1 & 0 & 0 \\
               0 & 1 & 0 \\
               0 & 0 & -1
      \end{array} \right) \; , & 
A_3 = \left( \begin{array}{ccc}
              -1 &  0 & 0 \\
               0 & -1 & 0 \\
               0 &  0 & 1
      \end{array} \right) \; , \nonumber \\ \\
B = \left( \begin{array}{ccc}
              1 & 0 & 0 \\
              0 & 0 & -1 \\
              0 & 1 & -1
    \end{array} \right) \; , & 
C = \left( \begin{array}{ccc}
              1 & 0 & 0 \\
              0 & 0 & -1 \\
              0 & 1 & 0
    \end{array} 