# Generation of Space Representation of non-crystallographic Point Groups

In Mathematica the command

FiniteGroupData[{"CrystallographicPointGroup",<group number>}, "SpaceRepresentation"]


yields the space representation (transformation matrices of the symmetry elements) of the 32 three-dimensional crystallographic point groups.

However if one e.g. looks a the symmetry of molecules, there are infinitely more Point groups which do not fulfill the Crystallographic Restriction Theorem with $C_5$, $D_7$ or $I_h$ as examples.

The underlying mechanism to generate a space representation of any group is the use of generators, finding a strong generating set and get the group elements out of it.

For permutation groups which are used as representation for point groups this can be done by the use of commands like GroupElements and GroupGenerators.

However I'm lost in how to use the very extensive group theory engine within Mathematica to generate the space representation of Point Groups out of the space representation of it's generators

$S_{10}= \begin{pmatrix} \frac{1}{4}(1+\sqrt{5}) & -\frac{1}{2}\sqrt{\frac{1}{2}(5-\sqrt{5})} & 0 \\ \frac{1}{2}\sqrt{\frac{1}{2}(5-\sqrt{5})} & \frac{1}{4}(1+\sqrt{5}) & 0 \\ 0 & 0 & -1 \end{pmatrix}$

and

$\sigma_d= \begin{pmatrix} -\frac{1+\sqrt{5}}{5+\sqrt{5}} & 0 & 2 \frac{1+\sqrt{5}}{5+\sqrt{5}} \\ 0 & 1 & 0 \\ 2 \frac{1+\sqrt{5}}{5+\sqrt{5}} & 0 & \frac{1+\sqrt{5}}{5+\sqrt{5}} \end{pmatrix}$

for $I_h$.

I hope this can be accomplished with on-board commands in Mathematica?

• Not an answer, but there is a way to get non-crystallographic point groups with the command DihedralGroup... e.g., DihedralGroup[7]//GroupElements gives you the permutation representation which can be then converted to real space. For other points groups, I guess one would first have to figure out if they're isomorphic to some DihedralGroup[n]. I guess that's a partial answer, but maybe not what you're looking for. – Jens Feb 5 '16 at 6:46
• @Jens: Thanks for the comment. How do you convert the permutation representation into real space? – Rainer Feb 5 '16 at 7:38
• I could do it for 2D (since we know where the polygon vertices are), but I think you're asking more generally about 3D... and I don't have a good general answer for that. – Jens Feb 5 '16 at 19:14