I am developing a library for geometric symmetry groups. The international tables of crystallography list the complete operator list for wallpaper groups and space groups, but I've been unable to locate such a list for 3D point groups. I've derived the seven axial groups myself here, but I haven't figured out the seven polygonal point groups yet. Does anyone have a list of all operators for $T$, $T_h$, $T_d$, $O$, $O_h$, $I$, and $I_h$? I would also prefer to follow existing conventions for the axes choices, if these have been standardized anywhere.
Expressing the operations in $(x,y,z)$ form is fine, although I will eventually need to translate that into 4D rotation matrix form.
I was unable to find any pre-existing operator list for 3D point groups, nor any standard for the choice of axis or order of operators. As such I ended up deriving the operators from their generators. The OpenSCAD code for the results should be easy to port to other languages if needed. Here's a summary:
Chiral tetrahedral ($T$)
Orientation: Given a cube oriented to the cartesian axes, we have 2-fold rotation axes through each face (i.e. along the axes) and 3-fold axes through each body diagonal.
Generators: 2-fold around X ($R_2$); 3-fold around (1,1,1) ($R_3$)
$$ R_2 = \begin{pmatrix} 1& 0& 0& 0\\ 0&-1& 0& 0\\ 0& 0&-1& 0\\ 0& 0& 0& 1 \end{pmatrix} $$ $$ R_3 = \begin{pmatrix} 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1 \end{pmatrix} $$
Operators (12):
$$ T = \{e, R_2, R_3, R_2 R_3, R_3 R_2, R_3 R_3, R_2 R_3 R_3, R_2 R_3 R_2, R_3 R_2 R_3, R_3 R_3 R_2, R_3 R_3 R_2 R_3, R_3 R_2 R_3 R_3 \} $$
Full tetrahedral ($T_d$)
Orientation: Same as $T$, with 6 additional mirror planes containing opposing edges of the cube.
Generators: Same as $T$, plus a reflection across x=y ($M_{x=y}$)
$$ M_{x=y} = \begin{pmatrix} 0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1 \end{pmatrix} $$
Operators (24):
All operators in $T$ plus mirrored versions $$ T_d = T \cup \left\{r\cdot M_{x=y}:r\in T\right\} $$
Pyritohedral ($T_h$)
Orientation: Same as $T$, with 6 additional mirror planes containing opposing edges of the cube.
Generators: Same as $T$, plus a reflection across xz ($M_{xz}$) $$ M_{xz} = \begin{pmatrix} 1& 0& 0& 0\\ 0&-1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1 \end{pmatrix} $$
Operators (24):
All operators in $T$ plus mirrored versions $$ T_h = T \cup \left\{r\cdot M_{xz}:r\in T\right\} $$
Chiral Octohedral ($O$)
Orientation: Given a cube oriented to the cartesian axes, we have 4-fold rotation axes through each face (i.e. along the axes), 3-fold axes through each body diagonal, and 2-fold rotations through the edge midpoints
Generators: 4-fold around X ($R_4$); 3-fold around (1,1,1) ($R_3$) $$ R_4 = \begin{pmatrix} 1& 0& 0& 0\\ 0& 0&-1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1 \end{pmatrix} $$
Operators (24):
$$ O = \{e, R_3, R_4, R_3 R_3, R_3 R_4, R_4 R_3, R_4 R_4, R_3 R_3 R_4, R_3 R_4 R_4, R_4 R_3 R_3, R_4 R_4 R_3, R_4 R_4 R_4, R_3 R_3 R_4 R_4, R_3 R_4 R_3 R_3, R_3 R_4 R_4 R_3, R_3 R_4 R_4 R_4, R_4 R_3 R_3 R_4, R_4 R_4 R_3 R_3, R_3 R_3 R_4 R_3 R_3, R_3 R_3 R_4 R_4 R_3, R_4 R_4 R_3 R_3 R_4, R_3 R_4 R_4 R_3 R_3, R_4 R_3 R_3 R_4 R_4, R_3 R_4 R_4 R_3 R_3 R_4\} $$
Full octohedral ($O_h$)
Orientation: Same as $O$, with additional mirror planes along the axes and containing opposing edges of the cube.
Generators: Same as $O$, plus a reflection across xz ($M_{xz}$)
Operators (48):
All operators in $O$ plus mirrored versions $$ O_h = O \cup \left\{r\cdot M_{xz}:r\in O\right\} $$
Chiral icosahedral ($I$)
Orientation: Orient an icosahedron such that each axis is along a 2-fold axis.
Generators: 2-fold around X ($R_2$), plus a 5-fold rotation around the $(\phi,0,1)$ axis ($R_5$).
$$ f = \frac{5-\sqrt{5}}{5+\sqrt{5}} \\ g = \frac{3+\sqrt{5}}{4} \\ \phi = \frac{1+\sqrt{5}}{2} \\ R_5 = \begin{pmatrix} 1-\frac{f}{2}& -\frac{\phi \sqrt{f}}{2}& \frac{\phi f}{2}& 0\\ \frac{\phi \sqrt{f}}{2}& 1-f (\frac{1}{2}+g)& -g \sqrt{f}&0\\ \frac{\phi f}{2}& g \sqrt{f}& 1-f g& 0\\ 0&0&0&1 \end{pmatrix} $$
Operators (60):
Use the decomposition $I = T\times C_5$:
$$ I = \left\{R_5^i \cdot r:r\in T, i \in [0\ldots4] \right\} $$
Full icosahedral ($I_h$)
Orientation: Same as $I$, with mirror planes
Generators: Same as $I$, plus XZ reflection ($M_{xz}$)
Operators (120):
$$ I_h = I \cup \left\{r \cdot M_{xz}:r\in I \right\} $$