Question: I have a particle which might have a c5, c6 or c7 symmetry. How can I find out which it is ?
Answer: Tricky. There is a possibility that the answer is "you can't". In most cases, however, it's possible to get a pretty accurate answer. In cases where the symmetry of the particle is unknown, the ability to distinguish between different symmetries is proportional to the overall contrast in the image. In cryo-EM there is always a tradeoff between contrast and resolution, so the best thing to do if you're trying to determine symmetry is exactly the opposite of what you'd do for high resolution. That is, take some micrographs in negative stain, or in ice, fairly far from focus at low voltage. This will provide the best overall contrast for an attempt to determine symmetry.
Once you've collected high-constrast data, there are a number of techniques to try to determine symmetry. for particles with a suspected Cn or Dn symmetry, startcsym is a good starting point. By running it several times with each possible symmetry you can see how well each one fits the data. Frequently comparing the symmetrized model in sym.hed with the class-averages in classes.hed will give the first indications of the true symmetry.
The next step is to try to refine each of the possible initial models and see if they 'fall apart' during refinement. This should resolve the symmetry question IF you have sufficient contrast, and IF your particles are in fairly random orientations. If the contrast is too low, or there is a strongly preferred orientation, however, an accurate answer may not be possible.
If the first technique fails, there are other possibilities, like using multivariate statistical analysis on an aligned set of raw particles. These issues are too complicated for discussuion in this FAQ.