I have not consulted your likns.

What type of X-ray data do you have? If you have single crystal X-ray data and have determined the unit cell parameters, then you know the crystal class apart from a few subtle cases of pseudosymmetry. Once you know the crystal class (cubic, tetragonal, orthorhombic etc.) then you have to find the lattice type, i.e. you to determine whether there is any centering of the cell. This is easy to find. You have to look for extinct reflections. Each centering has some characteristic extinction rule, viz. Face centered: reflections should have Miller indices hkl either all odd or all even, Body centered.: only reflection with h+k+l=2n (n integer), A face centered: k+l= 2n, B face centered: h+l=2n , C face centered h+k = 2n. If you have no centering then you have a primitive lattice P. Now open the space group table (International Tables of Crystallography A) and look for the space groups that your crystal may belong to. There will be in general quite a few for your crystal class. Now you have to find other symmetries of the lattice. For that you go to the table of space group extinction (See Internation Table or any textbook of X-ray crystallography like Staut and Jensen for example). You will find the space group extiction rules of each tabulated space group. You must compare X-ray data (mostly missing hkl) with the table and find the possible space groups. There may be a few possible cases. Sometimes you space group will be unique. You are then lucky. You may have often two or three choices that cannot be decided easily. Once you have a model of the structure and you have refined the intensity data assuming all these possible space groups and check the agreement factors then you will know more definitely to which space group your structure belongs to. If it is the question of deciding whether the space group is centrosymmetric or non-centrosymmetric (see previous discussion in RG) then the situation is more complex. You may have to do X-ray measurements with wavelength close to the absorption edge etc or use special methods like intensity statistics.

If you have powder diffraction data and have indexed all reflections successfully then you can use the above procedure for single crystal X-ray diffraction but the results may not be unambigous always. But trying several possible space groups to refine the structure (if you have a possible model) and looking carefully at the results and agreement factors you may be able to find uniquely the space group.

I have not consulted your likns.

What type of X-ray data do you have? If you have single crystal X-ray data and have determined the unit cell parameters, then you know the crystal class apart from a few subtle cases of pseudosymmetry. Once you know the crystal class (cubic, tetragonal, orthorhombic etc.) then you have to find the lattice type, i.e. you to determine whether there is any centering of the cell. This is easy to find. You have to look for extinct reflections. Each centering has some characteristic extinction rule, viz. Face centered: reflections should have Miller indices hkl either all odd or all even, Body centered.: only reflection with h+k+l=2n (n integer), A face centered: k+l= 2n, B face centered: h+l=2n , C face centered h+k = 2n. If you have no centering then you have a primitive lattice P. Now open the space group table (International Tables of Crystallography A) and look for the space groups that your crystal may belong to. There will be in general quite a few for your crystal class. Now you have to find other symmetries of the lattice. For that you go to the table of space group extinction (See Internation Table or any textbook of X-ray crystallography like Staut and Jensen for example). You will find the space group extiction rules of each tabulated space group. You must compare X-ray data (mostly missing hkl) with the table and find the possible space groups. There may be a few possible cases. Sometimes you space group will be unique. You are then lucky. You may have often two or three choices that cannot be decided easily. Once you have a model of the structure and you have refined the intensity data assuming all these possible space groups and check the agreement factors then you will know more definitely to which space group your structure belongs to. If it is the question of deciding whether the space group is centrosymmetric or non-centrosymmetric (see previous discussion in RG) then the situation is more complex. You may have to do X-ray measurements with wavelength close to the absorption edge etc or use special methods like intensity statistics.

If you have powder diffraction data and have indexed all reflections successfully then you can use the above procedure for single crystal X-ray diffraction but the results may not be unambigous always. But trying several possible space groups to refine the structure (if you have a possible model) and looking carefully at the results and agreement factors you may be able to find uniquely the space group.

Table 7.2 of the book Introduction to X-ray crystallography by M.M. Woolfson, p. 232 gives information about systematic absences for lattice types (P, A, B, C, F, I) and also for other symmetry elements like glide planes. The method of intensity statics has been discussed in the same book on p. 234. Sometime physical measurements like ferroelectricity and second harmonic generation yield information of space group type.

I see full explanation on the determination of the space group has been clearly given by Tapan Chatterji. Moreover, you can specify it more accurately with the help of some software, like like GSAS or X' Pert Plus.

Good luck

Please note that the systematic absences arise due to translational part of the symmetry; pure rotation, inversion or mirror do not give rise to absences. Glide planes and screw axes followed by translation (glide) produces absences. These are all tabulated in 7.2 of Woolfson's book, p. 232.

It is not a good idea to use any software without understanding how translation part of the symmetry gives rise to systematic absences. One should at least be able to write down the structure factor equation and then using the symmetry matrices check whether the structure factor becomes zero. If you cannot do it yourself then at least you try to understand the section 7.4 of Woolfson's book where the absences have been deterived for several symmetry elements involving translational part. Again please not that just lattice translations do not give rise to any systematic absences. I am refering to Woolfson's book because it is on my desk at home. You can very use any standard text book on X-ray crystallography. viz. Warren, Staut and Jensen, Buerger etc. etc. or International Table A.

General advice: Never use any software before you understand the basic principle.

Explanation given by Tapan is pretty good. You can match your data with the JCPDS data and find the space group.

The most complete information is available in International Tables for Crystallography A, section 3 "Space group determination and diffraction symbols". Table 3.2 gives reflection condition, diffraction symbols and possible space groups. This is the most complete available in the literature.

Since I know that most of you live with only X-ray or neutron powder diffraction data and single crystal data are not available to you, I shall give you (powder guys) some practical hints:

If only powder data are available then information on the direction of the reciprocal lattice is lost and the recognition of even the unit cell can be very difficult. The problem of determining the unit cell from a powder pattern, and indexing the diffraction peaks on the basis of the unit cell 9the indexing problem) is a very challenging one and is considered to be a bottle-neck problem. Once you have done this successfully and you are sure of the uniqueness of your solution (probably you are never with powder diffraction data) then you can proceed as I have indicated in above. Also you have to take care of the overlapping reflections that may introduce further difficulties.

First of all I must know what data you have. Let us suppose that you have a diffractogram from where you can know the 2theta and the corresponding intensity of the peaks. You should make a table of 2theta and intensity for various peaks, as well as the d-values calculated from the Bragg Law. As your sample is unknown, you should try from the cubic structure . In that case if supposing that you have a simple cubic lattice, all differeent combinations of hkl must be present. So you will be having a number of peaks. Try to find out what peak is 100. This will be corresponding to minimum value of d and all other values of d will be multiples of this value by an integer etc. Such procedures are there to actually indexing a sample. Please read carefully in "X-ray diffraction by B.D.Cullity" and you will be able to index any unknown sample.

On the other hand, computer programs are available to index unknown samples. Go to the web to find a suitable one for you.

There are standard procedures described in good x-ray crystallography books ( eg by M J Buerger) for the determination of space groups. The real issue is Friedel's law because of which the 230 space groups are represented by 122 diffraction symbols. Out of these 122, 50 symbols uniquely describe one space group each. For the remaining there are more than one possibilities for each diffraction symbol. The reason why there are more than one possibilities is because x-ray diffraction patterns cannot confirm or refute the existence or otherwise of the centre of symmetry. So corresponding to 230 space groups, one gets 122 diffraction symbols by adding a centre of symmetry to all the space groups. Remember, for the same reason, out of 32 point groups get clubbed into 11 Laue groups. One can use piezoelectricity, pyroelectricity, ferroelectricity etc to confirm the absence of centre of symmetry but the converse may lead to wrong conclusions if these effects are too weak too be measurable. Single crystal diffraction data can help you determine the diffraction symbols and Laue groups. After that, if you have more than one space group for the same diffraction symbol, you have to calculate the intensities and the one which gives values closest to observed intensities becomes the unique space group.

Link:

http://pd.chem.ucl.ac.uk/pdnn/symm4/condex.htm

You may try some softwares for initial guess like DICVOL6 and TREOR. These are free software and may be helpful for you.