[acd]

Do time crystal exist in higher dimensions than 3d since it has quantum properties? How would time crystals behave in those higher dimensions?

[posterboy]

I think it's fair to say that regular crystals exist in higher dimensions if they have more than 3 planes or axes of symmetry.

[hexane360]

This is definitely not true.

Here's the point group (rotational symmetries around a lattice point) for a typical cubic lattice found in many metals: http://materials.cmu.edu/degraef/pg/pg_mbar3m.gif

Here's the space group (rotational symmetries combined with lattice/translational symmetries): http://img.chem.ucl.ac.uk/sgp/large/225az1.htm This is for face-centered cubic crystals. That's 48 symmetry operations, for one of the most common crystal systems used. However, they don't exist in higher dimensions because they're not fully independent. You can construct the full symmetry from just a handful of operators.