How can I find the smaller symmetric structure from big crystal unit cell?
The structure contains 4 independent atoms, Mg, Al, Si and O. The space group is a cubic one containing 96 asymmetric units. If you place an atom on a general position, it will appear 96 times in the unit cell. Mg, Al, and Si are located in various special positions, i.e. lie on symmetry elements. Placing the Mg and the Si atom will result in 24 atoms in unit cell, while placing the Al atom in its position (on the origin) will result in 16 atoms in the unit cell (on a three-fold and on the center of inversion).

So you could say that the asymmetric unit contains one oxygen atom, one quarter Mg and Si atom respectively (24/96), and one sixth Al atom (16/96). One of the sites you cited in a previous question gives the fractional coordinates as:

Si: 0.3750 0.0000 0.2500

Al: Origin

Mg: 0.1250 0.0000 0.2500

O: 0.0341 0.0490 0.6530

If you look at the space group, http://img.chem.ucl.ac.uk/sgp/large/230az1.gif,



you can see that Al, Mg, and Si are on special positions (you can either find the locations on the diagram or apply the 96 operations (48 combined with the I-centering) listed on the right to the coordinates and see how many unique locations you get), while oxygen is not.


  can I manually cut this unit cell in 1/4 size, 40 atom system with 1/2 size for x, y, and z direction, and run the minimization?


If you software is aware of crystallographic symmetry, it is a very simple system with 4 degrees of freedom (the oxygen position and the lattice parameter a). Everything else is fixed if you impose the symmetry of garnet.


  I have Pyrope (Mg3Al2(SiO4)3) crystal structure, downloaded from materialsproject: https://materialsproject.org/materials/mp-6073/#. It appeared as triclinic cell...


There are some inconsistencies in the way the site shows the data. The space group is (correctly) given as Ia3d. The 3D image does not show a body-centered cubic cell, but instead the primitive rhombohedral cell (the smallest unit that recreates the entire crystal through three translations). In one iteration ("primitive"), the coordinate file lists the symmetry as 1, applied the Ia3d operators to the atomic positions (maybe to "not to confuse" users unaware of crystal symmetry), and switched to the rhombohedral cell.