Splitting of Equivalent Points in Noncentrosymmetric Space Groups Into Subsets Under Homogeneous Stress

Abstract

Splitting of general positions in crystals into subsets of equivalent sites under homogeneous stress has previously been given for all centrosymmetric space groups; the tabulation is here completed for all space groups by listing the results for noncentrosymmetric space groups.

1. Introduction

The present paper presents results analogous to those previously submitted for centrosymmetric space groups [1]¹ and for point groups [2]; these previous papers should be consulted for detailed discussion.

It is assumed that the symmetry elements possessed by a homogeneously stressed crystal will be those common to the crystal and to the macroscopic state of stress. Application of stress either leaves a space group unaltered or lowers it to a subgroup. Such lowering can always be considered to take place in successive steps each of which leaves no group which is both a stress-induced subgroup of the initial group and a supergroup of the final group, and which is distinct from both. Each such step can be accomplished by a uniaxial stress; for the noncentrosymmetric space groups all but two of the symmetry reductions consisting; of two or more successive steps can also be accomplished by uniaxial stress. These two require biaxial stress [2]. More general stress states are, however, consistent with many of the steps of symmetry lowering; we list the most general state of stress (in terms of the modified stress ellipsoid [1]) consistent with each step. The same stress is appropriate for all space-group-to-space-group transformations associated with a given point-group-to-point-group transformation. There are 25 of the latter which are minimum steps of symmetry lowering for noncentrosymmetric point groups so that the results for the noncentrosymmetric space groups are collected into 25 corresponding tables.

A set of points all of which are equivalent in the unstressed crystal frequently splits into two or more subsets under stress. For each space group all possibilities are taken into account by considering the behavior of the general position because the behavior of each special position can be derived by specializing the general position. This process of specialization in space groups has been discussed and a technique for visualizing it in terms of stereograms of point groups has been described [2].

2. Results

2.1. Behavior of General Position

The splitting of the general position (set of equivalent sites having no symmetry) into subsets is listed in tables 1 through 25. Each table is headed by a point-group transformation which is a minimum step of symmetry lowering. Each of the space groups associated with the initial point group is listed in the table together with the coordinates of a set of sites making up a general position. The latter are collected into subsets; all of the sites in a subset remain equivalent after symmetry reduction to the final space group which is also listed. For some of the point-group reductions the final point group can occur in two or three non-equivalent orientations. These may correspond to different final space groups; in table 12, for example, one orientation corresponds to the caption at the top of the table and the other to the caption at the bottom. The stress is described by giving conditions on the axes X, Y, Z of the stress ellipsoid [1] to the crystal axes x, y, z; the stress described is the most general (i.e., least lestricted) consistent with the symmetry reduction. In many of the tables a single stress specification suffices for all space groups, but in some (table 10, for example) the stress must be specified for each space group because it is customary to choose the axes in different orientations with respect to the point group.

Table 1.

Reduction from $\overline{4}3m$ to $\overline{4}2m$

Space group of unstrained crystal, order 24 per lattice point		If stressed so that X=Y; z∥Z			Space group of strained crystal, order 8 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	No.	Symbol
215	P$\overline{4}$3m	( x, y, z) ( y, x, z)	( y, z, x) ( z, y, x)	( z, x, y) ( x, z, y)	111	P$\overline{4}$2m
		( $\overline{y}$, x, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	( $\overline{z}$, y, $\overline{x}$) ( $\overline{y}$, z, $\overline{x}$)	( $\overline{x}$, z, $\overline{y}$) ( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z) ( $\overline{y}$, $\overline{x}$, z)	( $\overline{y}$, $\overline{z}$, x) ( $\overline{z}$, $\overline{y}$, x)	( $\overline{z}$, $\overline{x}$, y) ( $\overline{x}$, $\overline{z}$, y)
		( y, $\overline{x}$, $\overline{z}$) ( x, $\overline{y}$, $\overline{z}$)	( z, $\overline{y}$, $\overline{x}$) ( y, $\overline{z}$, $\overline{x}$)	( x, $\overline{z}$, $\overline{y}$) ( z, $\overline{x}$, $\overline{y}$)
216	F$\overline{4}$3m	( x, y, z) ( y, x, z)	( y, x, z) ( z, y, x)	( z, x, y) ( x, z, y)	119	I$\overline{4}$m2
		( $\overline{y}$, x, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	( $\overline{z}$, y, $\overline{x}$) ( $\overline{y}$, z, $\overline{x}$)	( $\overline{x}$, z, $\overline{y}$) ( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z) ( $\overline{y}$, $\overline{x}$, z)	( $\overline{y}$, $\overline{z}$, x) ( $\overline{z}$, $\overline{y}$, x)	( $\overline{z}$, $\overline{x}$, y) ( $\overline{x}$, $\overline{z}$, y)
		( y, $\overline{x}$, $\overline{z}$) ( x, $\overline{y}$, $\overline{z}$)	5 z, $\overline{y}$, $\overline{x}$) ( y, $\overline{z}$, $\overline{x}$)	( x, $\overline{z}$, $\overline{y}$) ( z, $\overline{x}$, $\overline{y}$)
217	I$\overline{4}$3m	( x, y, z) ( y, x, z)	( y, z, x) ( z, y, x)	( z, x, y) ( x, z, y)	121	I$\overline{4}$2m
		( $\overline{y}$, x, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	( $\overline{z}$, y, $\overline{x}$) ( $\overline{y}$, z, $\overline{x}$)	( $\overline{x}$, z, $\overline{y}$) ( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z) ( $\overline{y}$, $\overline{x}$, z)	( $\overline{y}$, $\overline{z}$, x) ( $\overline{z}$, $\overline{y}$, x)	( $\overline{z}$, $\overline{x}$, y) ( $\overline{x}$, $\overline{z}$, y)
		( y, $\overline{x}$, $\overline{z}$) ( x, $\overline{y}$, $\overline{z}$)	( z, $\overline{y}$, $\overline{x}$) ( y, $\overline{z}$, $\overline{x}$)	( x, $\overline{z}$, $\overline{y}$) ( z, $\overline{x}$, $\overline{y}$)
218	P$\overline{4}$3n	( x, y, z) (½+y,½+x,½+z)	( y, z, x) (½+z,½+y,½+x)	( z, x, y) (½+x,½+z,½+y)	112	P$\overline{4}$2c
		(½−y,½+x ½−z) ( $\overline{x}$, y, $\overline{z}$)	(½−z,½+y,½−x) ( $\overline{y}$, z, $\overline{x}$)	(½−x,½+z,½−y) ( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z) (½−y,½−x,½+z)	( $\overline{y}$, $\overline{z}$, x) (½−z,½−y,½+x)	( $\overline{z}$, $\overline{x}$, y) (½−x,½−z,½+y)
		(½+z,½−x,½−y) ( x, $\overline{y}$, $\overline{z}$)	(½+z,½−y,½−x) ( y, $\overline{z}$, $\overline{x}$)	(½+x,½−z,½−y) ( z, $\overline{x}$, $\overline{y}$)
219	F$\overline{4}$3c	( x, y, z) ( y, x,½+z)	( y, z, x) ( z, y,½+x)	( z, x, y) ( x, z,½+y)	120	I$\overline{4}$c2
		( $\overline{y}$, x,½−z) ( $\overline{x}$, y, $\overline{z}$)	( $\overline{z}$, y,½−x) ( $\overline{y}$. z, $\overline{x}$)	( $\overline{x}$, z,½−y) ( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z) ( $\overline{y}$. $\overline{x}$,½+z)	( $\overline{y}$, $\overline{z}$, x) ( $\overline{z}$, $\overline{y}$,½+x)	( $\overline{z}$, $\overline{x}$, y) ( $\overline{x}$, $\overline{z}$,½+y)
		( y, $\overline{x}$,½−z) ( x, $\overline{y}$, $\overline{z}$)	( z, $\overline{y}$,½−x) ( y, $\overline{z}$, $\overline{x}$)	( x, $\overline{z}$,½+y) ( z, $\overline{x}$, $\overline{y}$)
220	I$\overline{4}$3d	( x, y, z) (¼+y,¼+x,¼+z)	( y, z, x) (¼+z,¼+y,¼+x)	( z, x, y) (¼+x,¼+z,¼+y)	122	I$\overline{4}$2d
		(¾−y,¾+ x,¼− z) ( $\overline{x}$,½+y, ½−z)	(¾−z,¾+y,¼−x) ( $\overline{y}$,½+z,½−x)	(¾−x,¾+z,¼−y) ( $\overline{z}$,½+x,½−y)
		(½−x, $\overline{y}$,½+z) (¼−y, ¾−x,¾+z)	(½−y, $\overline{z}$,½+x) (¼−z,¾−y, ¾+x)	(½−z, $\overline{x}$,½+y) (¼−x,¾−z,¾+y)
		(¾+y, ¼−x, ¾−z) (½+x,½−y, $\overline{z}$)	(¾+z,¼−y,¾−x) (½+y,½−z, $\overline{x}$)	(¾+x,¼−z,¾−y) (½+z,½−x, $\overline{y}$)

Table 2.

Reduction from $\overline{4}3m$ to 3m

Space group of Wlstrained crystal, order 24 per lattice point		If stressed so that X=Y; [lll]∥Z				Space group of strained crystal, order 6 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	4th Subset	No.	Symbol
215	P$\overline{4}$3m	( x, y, z)	( x, $\overline{y}$, $\overline{x}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	160	R3m
		( y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		( z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		( y, x, z)	( $\overline{y}$, x, $\overline{z}$)	( y, $\overline{x}$, $\overline{z}$)	( $\overline{y}$, $\overline{x}$, z)
		( z, y, x)	( $\overline{z}$, $\overline{y}$, x)	( $\overline{z}$, y, $\overline{x}$)	( z, $\overline{y}$, $\overline{x}$)
		( x, z, y)	( x, $\overline{z}$, $\overline{y}$)	( $\overline{x}$, $\overline{z}$, y)	( $\overline{x}$, z, $\overline{y}$)
216	F$\overline{4}$3m	( x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$. y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	160	R3m
		( y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		( z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		( y, x, z)	( $\overline{y}$, x, $\overline{z}$)	( y, $\overline{x}$, $\overline{z}$)	( $\overline{y}$, $\overline{x}$, z)
		( z, y, x)	( $\overline{z}$, $\overline{y}$, x)	( $\overline{z}$, y, $\overline{x}$)	( z, $\overline{y}$, $\overline{x}$)
		( x, z, y)	( x, $\overline{z}$, $\overline{y}$)	( $\overline{x}$, $\overline{z}$, y)	( $\overline{x}$, z, $\overline{y}$)
217	I$\overline{4}$3m	( x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	160	R3m
		( y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		( z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		( y, x, z)	( $\overline{y}$, x, $\overline{z}$)	( y, $\overline{x}$, $\overline{z}$)	( $\overline{y}$, $\overline{x}$, z)
		( z, y, x)	( $\overline{z}$, $\overline{y}$, x)	( $\overline{z}$, y, $\overline{x}$)	( z, $\overline{y}$, $\overline{x}$)
		( x, z, y)	( x, $\overline{z}$, $\overline{y}$)	( $\overline{x}$, $\overline{z}$, y)	( $\overline{x}$, z, $\overline{y}$)
218	P$\overline{4}$3n	( x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	161	R3c
		( y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		( z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		(½+y, ½+x, ½+z)	(½−y, ½+x, ½−z)	(½+y, ½−x, ½−z)	(½−y, ½−x, ½+z)
		(½+z, ½+y, ½+x)	(½−z, ½−y, ½+x)	(½−z, ½+y, ½−x)	(½+z, ½−y, ½−x)
		(½+x, ½+z, ½+y)	(½+x, ½−z, ½−y)	(½−x, ½−z, ½+y)	(½−x, ½+z, ½−y)
219	F$\overline{4}$3c	( x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	161	R3c
		( y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		( z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		( y, x,½+z)	( $\overline{y}$, x,½−z)	( y, $\overline{x}$,½−z)	( $\overline{y}$, $\overline{x}$,½+z)
		( z, y, ½+x)	( $\overline{z}$, $\overline{y}$,½+x)	( $\overline{z}$, y,½−x)	( z, $\overline{y}$,½−x)
		( x, z,½+y)	( x, $\overline{z}$,½−y)	( $\overline{x}$.$\overline{z}$,½+y)	( $\overline{x}$, z,½−y)
220	I$\overline{4}$3d	( x, y, z)	(½+x,½−y, $\overline{z}$)	( $\overline{x}$,½+y,½−z)	(½−x, $\overline{y}$,½+z)	161	R3c
		( y, z, x)	(½−y, $\overline{z}$,½+x)	(½+y,½−z, $\overline{x}$)	( $\overline{y}$,½+z,½−x)
		( z, x, y)	( $\overline{z}$,½+x,½−y)	(½−z, $\overline{x}$,½+y)	(½+z,½−x, $\overline{y}$)
		(¼+y,¼+x,¼+z)	(¾−y,¾+x,¼−z)	(¾+y,¼−x,¾−z)	(¼−y,¾−x,¾+z)
		(¼+z,¼+y,¼+x)	(¼−z,¼−y,¾+x)	(¾−z,¾+y, ¼−x)	(¾+z,¼−y,¾−x)
		(¼+x,¼+z,¼+y)	(¾+x,¼−z,¾−y)	(¼−x,¾−z,¾+y)	(¾−x,¾+z,¼−y)

Table 3.

Reduction from 432 to 422

Space group of unstrained crystal, order 24 per lattice point		If stressed so that X=Y; z∥Z			Space group of strained crystal, order 8 per lattice point
		Coordinates referred to axes of unstrained crysta
No.	Symbol	1st Subset	2d Subset	3d Subset	No.	Symbol
207	P432	( x, y, z) ( y, x, $\overline{z}$)	( y, z, x) ( z, y, $\overline{x}$)	( z, x, y) ( x, z, $\overline{y}$)	89	P422
		( $\overline{y}$, x, z) ( $\overline{x}$, y, $\overline{z}$)	( $\overline{z}$, y, x) ( $\overline{y}$, z, $\overline{x}$)	( $\overline{x}$, z, y) ( y, x, z)
		( $\overline{x}$, $\overline{y}$, z) ( $\overline{y}$, $\overline{x}$, $\overline{z}$)	( $\overline{y}$, $\overline{z}$, x) ( $\overline{z}$, $\overline{y}$, $\overline{x}$)	( $\overline{z}$, $\overline{x}$, y) ( $\overline{x}$, $\overline{z}$, $\overline{y}$)
		( y, $\overline{x}$, z) ( x, $\overline{y}$, $\overline{z}$)	( z, $\overline{y}$, x) ( y, $\overline{z}$, $\overline{x}$)	( x, $\overline{z}$, y) ( z, $\overline{x}$, $\overline{y}$)
208	P4232	( x, y, z) (½+y,½+x,½−z)	( y, z, x) (½+z,½+y,½−x)	( z, x, y) (½+x,½+z,½−y)	93	P4222
		(½−y,½+x,½+z) ( $\overline{x}$, y, $\overline{z}$)	(½−z,½+y,½+x) ( $\overline{y}$. z, $\overline{x}$)	(½−x,½+z,½+y) ( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z) (½−y,½−x,½−z)	( $\overline{y}$, $\overline{z}$, x) (½−z,½−y,½−x)	( $\overline{z}$, $\overline{x}$, y) (½−x,½−z,½−y)
		(½+y,½−x,½+z) ( x, $\overline{y}$, $\overline{z}$)	(½+z,½−y,½+x) ( y, $\overline{z}$, $\overline{x}$)	(½+x,½−z,½+y) ( z, $\overline{x}$, $\overline{y}$)
209	F432	( x, y, z) ( y, x, $\overline{z}$)	( y, z, x) ( z, y, $\overline{x}$)	( z, x, y) ( x, z, $\overline{y}$)	97	I422
		( $\overline{y}$, x, z) ( $\overline{x}$, y, $\overline{z}$)	( $\overline{z}$, y, x) ( $\overline{y}$, z, $\overline{x}$)	( $\overline{x}$, z, y) ( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z) ( $\overline{y}$, $\overline{x}$, $\overline{z}$)	( $\overline{y}$, $\overline{z}$, x) ( $\overline{z}$, $\overline{y}$, $\overline{x}$)	( $\overline{z}$, $\overline{x}$, y) ( $\overline{x}$, $\overline{z}$, $\overline{y}$)
		( y, $\overline{x}$, z) ( x, $\overline{y}$, $\overline{z}$)	( z, $\overline{y}$, x) ( y, $\overline{z}$, $\overline{x}$)	( x, $\overline{z}$, y) ( z, $\overline{x}$, $\overline{y}$)
210	F4132	( x, y, z) (¼+y,¼+x,¼−z)	( y, z, x) (¼+z,¼+y,¼−x)	( z, x, y) (¼+x,¼+z,¼−y)	98	I4122
		(¼−y,¼+x,¼+z) ( $\overline{x}$, y, $\overline{z}$)	(¼−z,¼+y,¼+x) ( $\overline{y}$, z, $\overline{x}$)	(¼−x,¼+z,¼+y) ( $\overline{z}$, x, y)
		( $\overline{x}$, $\overline{y}$, z) (¼−y,¼−x,¼−z)	( $\overline{y}$, $\overline{z}$, x) (¼−z,¼−y,¼−x)	( $\overline{z}$, $\overline{x}$, y) (¼+x,¼−z,¼−y)
		(¼+y,¼−x,¼+z) ( x, $\overline{y}$, $\overline{z}$)	(¼+z,¼−y,¼+x) ( y, $\overline{z}$, $\overline{x}$)	(¼+x,¼−z,¼+y) ( z, $\overline{x}$, $\overline{y}$)
211	I432	( x, y, z) ( y, x, $\overline{z}$)	( y, z, x) ( z, y, $\overline{x}$)	( z, x, y) ( x, z, $\overline{y}$)	97	I422
		( $\overline{y}$, x, z) ( $\overline{x}$, y, $\overline{z}$)	( $\overline{z}$, y, x) ( y, x, z)	( $\overline{x}$, z, y) ( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z) ( $\overline{y}$, $\overline{x}$, $\overline{z}$)	( $\overline{y}$, $\overline{z}$, x) ( $\overline{z}$, $\overline{y}$, x)	( $\overline{z}$, $\overline{x}$, y) ( $\overline{x}$, $\overline{z}$, $\overline{y}$)
		( y, $\overline{x}$, z) ( x, $\overline{y}$, $\overline{z}$)	( z, $\overline{y}$, x) ( y, $\overline{z}$, $\overline{x}$)	( x, $\overline{z}$, y) ( z, $\overline{x}$, $\overline{y}$)
212	P4332	( x, y, z) (¼+y,¾+x,¾−z)	( y, z, x) (¼+z,¾+y,¾−x)	( z, x, y) (¼+x,¾+z,¾−y)	96	P43212
		(¾−y,¼+x,¾+z) ( $\overline{x}$,½+y,½−z)	(¾−z,¼+y,¾+x) ( $\overline{y}$,½+z,½−x)	(¾−x,¼+z,¾+y) ( $\overline{z}$,½+x,½−y)
		(½−x, $\overline{y}$,½+z) (¼−y,¼−x,¼−z)	(½−y, $\overline{z}$,½+x) (¼−z,¼−y,¼−x)	(½−z, $\overline{x}$,½+y) (¼−x,¼−z,¼−y)
		(¾+y,¾−x,¼+z) (½+x,½−y, $\overline{z}$)	(¾+z,¾−y,¼+x) (½+y,½−z, $\overline{x}$)	(¾+x,¾−z,¼+y) (½+z,½−x, $\overline{y}$)
213	P4132	( x, y, z) (¾+y,¾+x,¼−z)	( y, z, x) (¾+z,¼+y,¼−x)	( z, x, y) (¾+x,¼+ z,¼−y)	92	P41212
		(¼−y,¾+x,¼+z) ( $\overline{x}$,½+y,½−z)	(¼−z,¾+y,¼+x) ( $\overline{y}$,½+z,½−x)	(¼−x,¾+z,¼+y) ( $\overline{z}$,½+x,½−y)
		(½−x, $\overline{y}$,½+z) (¾−y,¾−x,¾−z)	(½−y, $\overline{z}$,½+x) (¾−z,¾−y,¾−x)	(½−z, $\overline{x}$,½+y) (¾−x,¾−z,¾−y)
		(¼+y,¼−x,¾+z) (½+x,½−y, $\overline{z}$)	(¼+z,¼−y,¾+x) (½+y,½−z, $\overline{x}$)	(¼+x,¼−z,¾+y) (½+z,½−x, $\overline{y}$)
214	I4132	( x, y, z) (¼+y,¾+x,¾−z)	( y, z, x) (¼+z,¾+y,¾−x)	( z, x, y) (¼+x,¾+z,¾−y)	98	I4122
		(¾−y,¼+x,¾+z) ( $\overline{x}$,½+y,½−z)	(¾−z,¼+y,¾+x) ( $\overline{y}$,½+z,½−x)	(¾−x,¼+z,¾+y) ( $\overline{z}$,½+x,½−y)
		(½−x, $\overline{y}$,½+z) (¼−y,¼−x,¼−z)	(½−y, $\overline{z}$,½+x) (¼−z,¼−y,¼−x)	(½−z, $\overline{x}$,½+y) (¼−x,¼−z,¼−y)
		(¾+y,¾−x,¼+z) (½+x,½−y, $\overline{z}$)	(¾+z,¾−y,¼+x) (½+y,½−z, $\overline{x}$)	(¾+x,¾−z,¼+y) (½+z,½−x, $\overline{y}$)

Table 4.

. Reduction from 432 to 32

Space group of unstrained crystal order 24 per lattice point		If stressed so that X=Y; [111] ∥ Z				Space group of strained crystal order 6 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	4th Subset	No.	Symbol
207	P432	( x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$ $\overline{y}$, z)	155	R32
		( $\overline{y}$, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		( z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		( $\overline{y}$, $\overline{x}$, $\overline{z}$)	( y, $\overline{x}$, z)	( $\overline{y}$, x, z)	( y, x, $\overline{z}$)
		( $\overline{z}$ , $\overline{y}$, $\overline{x}$)	( z, y, $\overline{x}$)	( z, $\overline{y}$, x)	( $\overline{z}$, y, x)
		( $\overline{x}$, $\overline{z}$, $\overline{y}$)	( $\overline{x}$, z, y)	( x, z, $\overline{y}$)	( x, $\overline{z}$, y)
208	P42,32	( x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	155	R32
		( y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		( z, x, y)	( $\overline{z}$, $\overline{x}$, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		(½-y,½-x,½-z)	(½+y,½-x,½+z)	(½-y,½-x,½+z)	(½+y,½+x,½-z)
		(½-z, ½-y, ½-x)	(½+z, ½+y, ½-x)	(½+z, ½-y, ½+x)	(½-z,½+y,½+x)
		(½-x, ½-z, ½-y)	(½-x,½+z, ½y)	(½+x, ½+z, ½-y)	(½+x, ½-z,½+y)
209	F432	( x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	155	R32
		( y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{y}$)
		( z, x, y)	( $\overline{z}$, x, y)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		( $\overline{y}$, $\overline{x}$, $\overline{z}$)	( y, $\overline{x}$, z)	( $\overline{y}$, x, z)	( y, x, $\overline{z}$)
		( $\overline{z}$, $\overline{y}$, $\overline{x}$)	( z, y, $\overline{x}$)	( z, $\overline{y}$, x)	( $\overline{z}$, y, x)
		( $\overline{x}$, $\overline{z}$, $\overline{y}$)	( $\overline{x}$, z, y)	( x, z, $\overline{y}$)	( x, $\overline{z}$, y)
210	F4132	( x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	150	1132
		( y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, x)
		( z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		(¼-y,¼-x¼,-z)	(¼+y,¼-x,¼+z)	(¼-y, ¼+x, ¼+z)	(¼+y, ¼+x, ¼-z)
		(¼-z, ¼-y, ¼-x)	(¼+z, ¼+y, ¼-x)	(¼+z, ¼-y, ¼+x)	(¼-z, ¼+y, ¼+x)
		(¼-x, ¼-z, ¼-y)	(¼-x, ¼+z, ¼+y)	(¼+x, ¼ +z, ¼-y)	(¼+x, ¼-z, ¼+y)
211	I432	( x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	155	R32
		( y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		( z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
		( $\overline{y}$, $\overline{x}$, $\overline{z}$)	( y, $\overline{x}$, z)	( $\overline{y}$, x, z)	( y, x, $\overline{z}$)
		( $\overline{z}$, $\overline{y}$, $\overline{x}$)	( z, y, $\overline{x}$)	( z, $\overline{y}$ , x)	( $\overline{z}$, y, x)
		( $\overline{x}$, $\overline{z}$, $\overline{y}$)	( $\overline{x}$, z, y)	( x, z, $\overline{y}$)	( x, $\overline{z}$, y)
212	P4332	( x, y, z)	(½+z, ½-y, $\overline{z}$)	( $\overline{x}$,½+y,½-z)	(½,-x, $\overline{y}$, ½+z)	155	R32
		( y, z, x)	(½-y, $\overline{z}$,½+x)	( +y, ½-z, $\overline{x}$)	( $\overline{y}$, ½+z, ½-x)
		( z, x, y)	( $\overline{z}$, ½+x,½-y)	(½-z, $\overline{x}$, ½+y)	(½+z, ½-x, $\overline{y}$)
		(¼-y, ¼-x, ¼-z)	(¾+y,¾-x,¼+z)	(¾-y,¼+x,¾+z)	(¼+y,¾+x,¾-z)
		(¼-z,¼-y,¼-x)	(¼+z, ¾+y, ¾-x)	(¾+z,¾-y,¼+x)	(¾-z,¼+y, ¾+x)
		(¼-x,¼ -z, ¼-y)	(¾-x,¼+z,¾+y)	(¼+x,¾+z,¾-y)	(¾+x,¾-z,¼+V)
213	P4132	( x, y, z)	(½-x, ½-y, $\overline{z}$)	( $\overline{x}$,½+y,½-z)	(½,-x, $\overline{y}$, ½+z)	155	K32
		( y, z, x)	(½-y, $\overline{z}$, ½-x)	(½+y,½-z, $\overline{x}$)	( $\overline{y}$, +z, ½-x)
		( z, x, y)	( $\overline{z}$, ½+x, ½-y)	(½-z, $\overline{x}$,½-y)	(½+z, ½-x, $\overline{y}$)
		(¾-y,¾-x,¾-z)	(¼+y, ¼-x, ¼+z)	(¼-y,¾+x,¼+z)	(¾+y,¼+x,¼-z)
		(¾-z,¾-y,¾-x)	(¾+z,¼+y,¼-x)	(¼+z, ¼-y, ¾+x)	(¼-z,¾+y,¼+x)
		(¾-x, ¾-z, ¾-y)	(¼-x, ¾+z, ¼+y)	(¾+x, ¼+z,¼-y)	(¼+x,¼ -z,¾+y)
214	I4132	( x, y, z)	(½+x, ½-y, $\overline{z}$)	( $\overline{x}$,½+y,½-z)	(½-x, $\overline{y}$,½+z)	155	R32
		( y, z, x)	(½-y, $\overline{z}$, ½+x)	(½+y,½-z, $\overline{x}$)	( $\overline{y}$,½+z,½-x)
		( z, x, y)	( $\overline{z}$, ½+x, ½-y)	(¾-z, $\overline{x}$,½+y)	(½+z,½-x, $\overline{y}$)
		(¼-y,¼-x,¼-z)	(¾+y, ¾-x, ¼+z)	(¾-y,¼+x,¾+z)	(¼+y,¾+x,¾-z)
		(¼-z,¼-y,¼-x)	(¼+z, ¾+y, ¾-x)	(¾+z, ¾-y,¼+ x)	(¾-z,+,¼-y,¾+x)
		(¼-x,¼-z,¼-v)	(¾-x, ¼+z,¾+y)	(¼+x,¾+z,¾-y)	(¾+x,¾-z,¼+y)

Table 5.

Reduction from 23 to 222

Space group of unstrained crystal, order 12 per lattice point		If stressed so that x, y, z ∥X, Y, Z, any permutation			Space group of strained crystal, order 4 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	No.	Symbol
195	P23	( x, y, z)	( y, z, x)	( z, x, y)	16	P222
		( x, $\overline{y}$, $\overline{z}$)	( y, $\overline{z}$, $\overline{x}$)	( z, $\overline{x}$, $\overline{y}$)
		( $\overline{x}$, y, $\overline{z}$)	( $\overline{y}$, z, $\overline{x}$)	( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z)	( $\overline{y}$, $\overline{z}$, x)	( $\overline{z}$, $\overline{x}$, y)
196	F23	( x, y, z)	( y, z, x)	( z, x, y)	22	F222
		( x, $\overline{y}$, $\overline{z}$)	( y, $\overline{z}$, $\overline{x}$)	( z, $\overline{x}$, $\overline{y}$)
		( $\overline{x}$, y, $\overline{z}$)	( $\overline{y}$, z, $\overline{x}$)	( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z)	( $\overline{y}$, $\overline{z}$, x)	( $\overline{z}$, $\overline{x}$, y)
197	I23	( x, y, z)	( y, z, x)	( z, x, y)	23	I222
		( x, $\overline{y}$, $\overline{z}$)	( y, $\overline{z}$, $\overline{x}$)	( z, $\overline{x}$, $\overline{y}$)
		( $\overline{x}$, y, $\overline{z}$)	( $\overline{y}$, z, $\overline{x}$)	( $\overline{z}$, x, $\overline{y}$)
		( $\overline{x}$, $\overline{y}$, z)	( $\overline{y}$, $\overline{z}$, x)	( $\overline{z}$, $\overline{x}$, y)
198	P213	( x, y, z)	( y, z, x)	( z, x, y)	19	P212121
		(½+x, ½−y, $\overline{z}$)	(½+y, ½−z, $\overline{x}$)	(½+z, ½−x, $\overline{y}$)
		( $\overline{x}$, ½+y, ½−z)	( $\overline{y}$, ½+z, ½−x)	( $\overline{z}$, ½+x, ½−y)
		(½−x, $\overline{y}$, ½+z)	(½−y, $\overline{z}$, ½+x)	(½−z, $\overline{x}$, ½+y)
199	1213	( x, y, z)	( y, z, x)	( z, x, y)	24	I212121
		(½+x, ½−y, $\overline{z}$)	(½+y, ½−z, $\overline{x}$)	(½+z, ½−x, $\overline{y}$)
		( $\overline{x}$, ½+y, ½−z)	( $\overline{y}$, ½+z, ½−x)	( $\overline{z}$, ½+x, ½−y)
		(½−x, $\overline{y}$, ½+z)	(½−y, $\overline{z}$, ½+x)	(½−z, $\overline{x}$, ½+y)

Table 6.

Reduction from 23 to 3

Space group of unstrained crystal, order 12 per lattice point		If stressed so that X = Y; [111] ∥Z				Space group of strained crystal, order 3 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	4th Subset	No.	Symbol
195	P23	(x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	146	R3
		(y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		(z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
196	F23	(x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	146	R3
		(y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		(z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
197	I23	(x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$)	( $\overline{x}$, $\overline{y}$, z)	146	R3
		(y, z, x)	( $\overline{y}$, $\overline{z}$, x)	( y, $\overline{z}$, $\overline{x}$)	( $\overline{y}$, z, $\overline{x}$)
		(z, x, y)	( $\overline{z}$, x, $\overline{y}$)	( $\overline{z}$, $\overline{x}$, y)	( z, $\overline{x}$, $\overline{y}$)
198	P213	(x, y, z)	(½+x, ½−y, $\overline{z}$)	( $\overline{x}$, ½+y, ½−z)	(½−x, $\overline{y}$, ½+z)	146	R3
		(y, z, x)	(½−y, $\overline{z}$, ½+x)	(½+y, ½−z, $\overline{x}$)	( $\overline{y}$, ½+z, ½−x)
		(z, x, y)	( $\overline{z}$, ½+x, ½−y)	(½−z, $\overline{x}$, ½+y)	(½+z, ½−x, $\overline{y}$)
199	I213	(x, y, z)	(½+x, ½−y, $\overline{z}$)	( $\overline{x}$, ½+y, ½−z)	(½−x, $\overline{y}$, ½+z)	146	R3
		(y, z, x)	(½−y, $\overline{z}$, ½+x)	(½+y, ½−z, $\overline{x}$)	( $\overline{y}$, ½+z, ½−x)
		(z, x, y)	( $\overline{z}$, ½+x, ½−y)	(½−z, $\overline{x}$, ½+y)	(½+z, ½−x, $\overline{y}$)

Table 7.

Reduction from $\overline{6}m2$ to mm2

Space group of unstrained crystal, order 12 per lattice point		If stressed so that y and z ∥ any two of X, Y, and Z			Space group of strained crystal, order 4 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	No.	Symbol
187	P$\overline{6}$m2	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	38	Amm2
		(x,x—y, z)	($\overline{y}$, $\overline{x}$, z)	(y-x,y, z)
		(x, y, $\overline{z}$)	($\overline{y}$,x-y, $\overline{z}$)	(y-x,$\overline{x}$, $\overline{z}$)
		(x,x—y, $\overline{z}$)	($\overline{y}$, $\overline{x}$, $\overline{z}$)	(y-x,y, $\overline{z}$)
188	P$\overline{6}$c2	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	40	Ama2
		(x,x-y,½-z)	(y, $\overline{x}$, ½-z)	(y-x,y,½-z)
		(x, y, ½-z)	($\overline{y}$,x-Y,½-z)	(y-x,$\overline{x}$,½-z)
		(x,x— y, $\overline{z}$)	($\overline{y}$, $\overline{x}$, $\overline{z}$)	(y-x,y, $\overline{z}$)
189	P$\overline{6}$2m	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	38	Amm2
		(x, y-x, z)	(y, x, z)	(x-y, y, z)
		(x, y, $\overline{z}$)	($\overline{y}$,x-y, $\overline{z}$)	(y-x,$\overline{x}$, $\overline{z}$)
		($\overline{x}$,y-x, $\overline{z}$)	(y, x, $\overline{z}$)	(x-y,$\overline{y}$, $\overline{z}$)
190	P$\overline{6}$2e	(x, y, z)	($\overline{y}$, x-y, z)	(y-x,$\overline{x}$, z)	40	Ama2
		($\overline{x}$,y-x,½+z)	(y, x, ½+z)	(x-y, y, ½+z)
		(x, y, ½-z)	($\overline{y}$,x-y, ½-z)	(y-x,$\overline{x}$,½-z)
		(x,y-x, $\overline{z}$)	(y, x, $\overline{z}$)	(x-y,$\overline{y}$, $\overline{z}$)

Table 8.

Reduction from 6mm to mm2

Space group of unstraincd crystal, order 12 per lattice point		If stressed so that y and z ∥ any two of X, Y, and Z			Space group of strained crystal, order 4 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	No.	Symbol
183	P6mm	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	35	Cmm2
		(x,x-y, z)	($\overline{y}$, $\overline{x}$, z)	(y-x,y, z)
		($\overline{x}$, $\overline{y}$, z)	(y,y-x, z)	(x-y,x, z)
		($\overline{x}$,y—x, z)	(y, x, z)	(x-y,$\overline{y}$, z)
184	P6cc	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	37	Ccc2
		(x,x-y,½+z)	($\overline{y}$, $\overline{x}$,½+z)	(y-x,y,½+z)
		($\overline{x}$, $\overline{y}$, z)	(y,y-x, z)	(x-y,x, z)
		($\overline{x}$,y—x,½+z)	(y, x,½+z)	(x-y,$\overline{y}$,½+z)
185	P63cm	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	36	Cmc21
		(x,x-y,½+z)	($\overline{y}$, $\overline{x}$,½+z)	(y-x,y,½+z)
		($\overline{x}$, $\overline{y}$,½+z)	(y,y-x,½+z)	(x-y,x,½+z)
		($\overline{x}$,y—x, z)	(y, x, z)	(x-y,$\overline{y}$, z)
186	P63mc	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	36	Cmc21
		(x,x—y, z)	($\overline{y}$, $\overline{x}$, z)	(y-x,y, z)
		($\overline{x}$, $\overline{y}$,½+z)	(y,y-x,½+z)	(x-y,x,½+z)
		($\overline{x}$,y—x,½+z)	(y, x,½+z)	(x-y,$\overline{y}$,½+z)

Table 9.

Reduction from 622 to 222

Space group of unstrained crystal, order 12 per lattice point		If stressed so that y and z ∥ any two of X, Y, and Z			Space group of strained crystal, order 4 per lattice point
		Coordinates referred to axes of unstrained crysta
No.	Symbol	1st Subset	2d Subset	3d Subset	No.	Symbol
177	P622	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	21	C222
		(x,x-y, $\overline{z}$)	($\overline{y}$, $\overline{x}$, $\overline{z}$)	(y-x,y, $\overline{z}$)
		($\overline{x}$, $\overline{y}$, z)	(y, y -x, z)	(x-y,x, z)
		($\overline{x}$, y-x, $\overline{z}$)	(y, x, $\overline{z}$)	(x- y, $\overline{y}$, $\overline{z}$)
178	P6122	(x, y, z)	($\overline{y}$,x-y,⅓+z)	(y-x,$\overline{x}$,⅔+z)	20	C2221
		(x,x-y,⅙-z)	($\overline{y}$, $\overline{x}$,⅚-z)	(y-x,y,½-z)
		($\overline{x}$, $\overline{y}$,½+z)	(y, y -x,⅚+z)	(x-y,x,⅙+z))
		($\overline{x}$, y-x,⅔-z)	(y x,⅓-z)	(x- y, $\overline{y}$, $\overline{z}$)
179	P6522	(x, y, z)	($\overline{y}$,x-y,⅔+z)	(y-x,$\overline{x}$,⅓+z)	20	C2221
		(x,x-y,⅚-z)	($\overline{y}$, $\overline{x}$,⅙-z)	(y-x,y,½-z)
		($\overline{x}$, $\overline{y}$,½+z)	(y, y -x,⅙+z)	(x-y,x,⅚+z)
		($\overline{x}$, y-x,⅓-z)	(y x,⅔-z)	(x- y, $\overline{y}$, $\overline{z}$)
180	P6222	(x, y, z)	($\overline{y}$,x-y,⅔+z)	(y-x,$\overline{x}$,⅓+z)	21	C222
		(x,x-y,⅓-z)	($\overline{y}$, $\overline{x}$,⅔-z)	(y-x,y, $\overline{z}$)
		($\overline{x}$, $\overline{y}$, z)	(y, y -x,⅔+z)	(x-y,x,⅓+z)
		($\overline{x}$, y-x,⅓-z)	(y x,⅔-z)	(x- y, $\overline{y}$, $\overline{z}$)
181	P6422	(x, y, z)	($\overline{y}$,x-y,⅓+z)	(y-x,$\overline{x}$,⅔+z)	21	C222
		(x,x-y,⅔-z)	($\overline{y}$, $\overline{x}$,⅓-z)	(y-x,y, $\overline{z}$)
		($\overline{x}$, $\overline{y}$, z)	(y, y -x,⅓+z)	(x-y,x,⅔+z)
		($\overline{x}$, y-x,⅔-z)	(y x,⅓-z)	(x- y, $\overline{y}$, $\overline{z}$)
182	P6322	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	20	C2221
		(x,x-y,½-z)	($\overline{y}$, $\overline{x}$,½-z)	(y-x,y,½-z)
		($\overline{x}$, $\overline{y}$,½+z)	(y, y -x,½+z)	(x-y,x,½+z)
		($\overline{x}$, y-x, $\overline{z}$)	(y, x, $\overline{z}$)	(x- y, $\overline{y}$, $\overline{z}$)

Table 10.

Reduction from $\overline{4}2m$ to mm2

Space group of unstrained crystal, order 8 per lattice point		If stressed so that	Coordinates referred to axes of unstrained crystal				Space group of strained crystal, order 4 per lattice point
No.	Symbol		1st Subset		2d Subset		No.	Symbol
111	P$\overline{4}$2m	[1$\overline{1}$0], [110],z ∥ X, Y, Z, any permutation.	(x, y, z)	( y, x, z)	( x, $\overline{y}$, $\overline{z}$)	(y, $\overline{x}$, $\overline{z}$)	35	Cmm2
			($\overline{x}$, $\overline{y}$, z)	($\overline{y}$, $\overline{x}$, z)	( $\overline{x}$, y, $\overline{z}$)	($\overline{y}$, x, $\overline{z}$)
112	P$\overline{4}$2c	Do.	(x, y, z)	( y, x,½+ z)	( x, $\overline{y}$,½-z)	(y, $\overline{x}$, $\overline{z}$)	37	Ccc2
			($\overline{x}$, $\overline{y}$, z)	($\overline{y}$, $\overline{x}$,½+z)	( $\overline{x}$, y,½-z)	($\overline{y}$, x, $\overline{z}$)
113	P$\overline{4}$21m	Do,	(x, y, z)	(½+y,½+x, z)	(½+x,½-y, $\overline{z}$)	(y, $\overline{x}$, $\overline{z}$)	35	Cmm2
			($\overline{x}$, $\overline{y}$, z)	(½-y,½-x, z)	(½-x,½+y, $\overline{z}$)	($\overline{y}$, x, $\overline{z}$)
114	P$\overline{4}$21c	Do,	(x, y, z)	(½+y,½+x,½+ z)	(½+x,½-y,½-z)	(y, $\overline{x}$, $\overline{z}$)	37	Ccc2
			($\overline{x}$, $\overline{y}$, z)	(½-y,½-x,½+z)	(½-x,½+y,½-z)	($\overline{y}$, x, $\overline{z}$)
115	P$\overline{4}$m2	x, y, z ∥. X, Y, Z, any permutation.	(x, y, z)	( x, $\overline{y}$, z)	( y, x, $\overline{z}$)	(y, $\overline{x}$, $\overline{z}$)	25	Pmm2
			($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y, z)	($\overline{y}$, $\overline{x}$, $\overline{z}$)	($\overline{y}$, x, $\overline{z}$)
116	P$\overline{4}$c2	Do.	(x, y, z)	( x, $\overline{y}$,½+z)	( y, x,½-z)	(y, $\overline{x}$, $\overline{z}$)	27	Pcc2
			($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y,½+z)	($\overline{y}$, $\overline{x}$,½-z)	($\overline{y}$, x, $\overline{z}$)
117	P4b2	Do.	(x, y, z)	(½+x,½-y, z)	(½+y,½+x, $\overline{z}$)	(y, $\overline{x}$, $\overline{z}$)	32	Pba2
			($\overline{x}$, $\overline{y}$, z)	(½-x,½-y, z)	(½-y,½-x, $\overline{z}$)	($\overline{y}$, x, $\overline{z}$)
118	P$\overline{4}$n2	Do.	(x, y, z)	(½+x,½-y,½+z)	(½+y,½+x,½-z)	(y, $\overline{x}$, $\overline{z}$)	34	Pnn2
			($\overline{x}$, $\overline{y}$, z)	(½-x,½+y,½+z)	(½-y,½-x,½-z)	($\overline{y}$, x, $\overline{z}$)
119	I$\overline{4}$m2	Do.	(x, y, z)	( x, $\overline{y}$, z)	( y, x, $\overline{z}$)	(y, $\overline{x}$, $\overline{z}$)	44	Imm2
			($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y, z)	($\overline{y}$, $\overline{x}$, $\overline{z}$)	($\overline{y}$, x, $\overline{z}$)
120	I$\overline{4}$c2	Do.	(x, y, z)	( x, $\overline{y}$,½+z)	( y, x,½-z)	(y, $\overline{x}$, $\overline{z}$)	45	Iba2
			($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y,½+z)	($\overline{y}$, $\overline{x}$,½-z)	($\overline{y}$, x, $\overline{z}$)
121	I$\overline{4}$2m	[1$\overline{1}$0]. [110], z ∥ X, Y, Z, any permutation.	(x, y, z)	( y, x, z)	( x, $\overline{y}$, $\overline{z}$)	(y, $\overline{x}$, $\overline{z}$)	42	Fmm2
			($\overline{x}$, $\overline{y}$, z)	($\overline{y}$, $\overline{x}$, z)	( $\overline{x}$, y, $\overline{z}$)	($\overline{y}$, x, $\overline{z}$)
122	I$\overline{4}$2d	Do.	(x, y, z)	( y,½+x,¼+ z)	( x,½-y,¼-z)	(y, $\overline{x}$, $\overline{z}$)	43	Fdd2
			($\overline{x}$, $\overline{y}$, z)	( $\overline{y}$,½-x,¼+ z)	( $\overline{x}$,½+y,¼-$\overline{z}$)	($\overline{y}$, x, $\overline{z}$)

Table 11.

Reduction from $\overline{4}2m$ to 222

Space group of unstrained crystal, order 8 per lattice point		If stressed so that	Coordinates referred to axes of unstrained crystal				Space group of strained crystal, order 4 per lattice point
No.	Symbol		1st Subset		2d Subset		No.	Symbol
111	P$\overline{4}$2m	x, y, z II X, Y, Z, any permutation.	(x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( y, x, z)	(y, $\overline{x}$, $\overline{z}$)	16	P222
			($\overline{x}$, $\overline{y}$, z)	($\overline{x}$, y, $\overline{z}$)	( $\overline{y}$, $\overline{x}$, z)	($\overline{y}$, x, $\overline{z}$)
112	P$\overline{4}$2c	Do.	(x, y, z)	( x, $\overline{y}$,½-z)	( y, x,½+z)	(y, $\overline{x}$, $\overline{z}$)	16	P222
			($\overline{x}$, $\overline{y}$, z)	($\overline{x}$, y,½-z)	( $\overline{y}$, $\overline{x}$,½+z)	($\overline{y}$, x, $\overline{z}$)
113	P$\overline{4}$21m	Do.	(x, y, z)	(½+x,½-y, $\overline{z}$)	(½+y,½+x, z)	(y, $\overline{x}$, $\overline{z}$)	18	P21212
			($\overline{x}$, $\overline{y}$, z)	(½-x,½+y, $\overline{z}$)	(½-y,½-x, z)	($\overline{y}$, x, $\overline{z}$)
114	P$\overline{4}$21C	Do.	(x, y, z)	(½+x,½-y,½-z)	(½+y,½+x,½+z)	(y, $\overline{x}$, $\overline{z}$)	18	P21212
			($\overline{x}$, $\overline{y}$, z)	(½-x,½+y,½-z)	(½-y,½-x,½+z)	($\overline{y}$, x, $\overline{z}$)
115	P$\overline{4}$m2	[1$\overline{1}$0], [110], z ∥ X, Y, Z, any permutation.	(x, y, z)	( y, x, $\overline{z}$)	( x, $\overline{y}$, z)	(y, $\overline{x}$, $\overline{z}$)	21	C222
			($\overline{x}$, $\overline{y}$, z)	( $\overline{y}$, $\overline{x}$, $\overline{z}$)	($\overline{x}$, y, z)	($\overline{y}$, x, $\overline{z}$)
116	P$\overline{4}$c2	Do.	(x, y, z)	( y, x,½-z)	( x, $\overline{y}$,½+z)	(y, $\overline{x}$, $\overline{z}$)	21	C222
			($\overline{x}$, $\overline{y}$, z)	( $\overline{y}$, $\overline{x}$,½-z)	($\overline{x}$, y,½+z)	($\overline{y}$, x, $\overline{z}$)
117	P$\overline{4}$b2	Do.	(x, y, z)	( ½+y,½+x, $\overline{z}$)	(½+x,½-y, z)	(y, $\overline{x}$, $\overline{z}$)	21	C222
			($\overline{x}$, $\overline{y}$, z)	( ½-y,½-x, $\overline{z}$)	(½-x,½-y, z)	($\overline{y}$, x, $\overline{z}$)
118	P$\overline{4}$n2	Do.	(x, y, z)	( ½+y,½+x,½-z)	(½+x,½-y,½+z)	(y, $\overline{x}$, $\overline{z}$)	21	C222
			($\overline{x}$, $\overline{y}$, z)	( ½-y,½-x,½-z)	(½-x,½+y,½+z)	($\overline{y}$, x, $\overline{z}$)
119	1$\overline{4}$m2	Do.	(x, y, z)	( y, x, $\overline{z}$)	( x, $\overline{y}$, z)	(y, $\overline{x}$, $\overline{z}$)	22	F222
			($\overline{x}$, $\overline{y}$, z)	( $\overline{y}$, $\overline{x}$, $\overline{z}$)	($\overline{x}$, y, z)	($\overline{y}$, x, $\overline{z}$)
120	1$\overline{4}$c2	Do.	(x, y, z)	( y, x,½-z)	( x, $\overline{y}$,½+z)	(y, $\overline{x}$, $\overline{z}$)	22	F222
			($\overline{x}$, $\overline{y}$, z)	( $\overline{y}$, $\overline{x}$,½-z)	($\overline{x}$, y,½+z)	($\overline{y}$, x, $\overline{z}$)
121	I$\overline{4}$2m	x, y, z ∥ X, Y, Z, any permutation.	(x, y, z)	( x, $\overline{y}$, $\overline{z}$)	( y, x, z)	(y, $\overline{x}$, $\overline{z}$)	23	I222
			($\overline{x}$, $\overline{y}$, z)	($\overline{x}$, y, $\overline{z}$)	( $\overline{y}$, $\overline{x}$, z)	($\overline{y}$, x, $\overline{z}$)
122	I$\overline{4}$2d	Do.	(x, y, z)	( x,½-y,¼-z)	( y,½+x,¼+z)	(y, $\overline{x}$, $\overline{z}$)	24	I212121
			($\overline{x}$, $\overline{y}$, z)	($\overline{x}$,½-y,¼-z)	($\overline{y}$,½-x,¼+z)	($\overline{y}$, x, $\overline{z}$)

Table 12.

Reduction from 4mm to mm2

Space group of unstrained crystal, order 8 per lattice point		If stressed so that x, y, z ∥ X, Y, Z, any permutation				Space group of strained crystal, order 4 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset		2d Subset		No.	Symbol
99	P4mm	(x, y, z)	( x, $\overline{y}$, z)	( y, x, z)	( y, $\overline{x}$, z)	25	Pmm2	35	Cmm2
		($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y, z)	( $\overline{y}$, $\overline{x}$, z)	( $\overline{y}$, x, z)
100	P4bm	(x, y, z)	(½+x,½-y, z)	(½+y,½+x, z)	( y, $\overline{x}$, z)	32	Pba2	35	Cmm2
		($\overline{x}$, $\overline{y}$, z)	(½-x,½+y, z)	(½-y,½-x, z)	( $\overline{y}$, x, z)
101	P42cm	(x, y, z)	( x, $\overline{y}$,½+z)	( y, x, z)	( y, $\overline{x}$,½+z)	27	Pcc2	35	Cmm2
		($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y,½+z)	( $\overline{y}$, $\overline{x}$, z)	( $\overline{y}$, x,½+z)
102	P42nm	(x, y, z)	(½+x,½-y,½+z)	( y, x, z)	(½+y,½-x,½+z)	34	Pnn2	35	Cmm2
		($\overline{x}$, $\overline{y}$, z)	(½-x,½-y,½+z)	( $\overline{y}$, $\overline{x}$, z)	(½-y,½+x,½+z)
103	P4cc	(x, y, z)	( x, $\overline{y}$,½+z)	( y, x,½+z)	( y, $\overline{x}$, z)	27	Pcc2	37	Ccc2
		($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y,½+z)	( $\overline{y}$, $\overline{x}$,½+z)	( $\overline{y}$, x, z)
104	P4nc	(x, y, z)	(½+x,½-y,½+z)	(½+y,½+x,½+z)	( $\overline{y}$, x,½+z)	31	Pnn2	37	Ccc2
		($\overline{x}$, $\overline{y}$, z)	(½-x,½-y,½+z)	(½-y,½-x,½+z)	( $\overline{y}$, x, z)
105	P42mc	(x, y, z)	( x, $\overline{y}$, z)	( y, x,½+z)	( y, $\overline{x}$,½+z)	25	Pmm2	37	Ccc2
		($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y, z)	( $\overline{y}$, $\overline{x}$,½+z)	( $\overline{y}$, x,½+z)
106	P42bc	(x, y, z)	(½+x,½-y, z)	(½+y,½+x,½+z)	( y, $\overline{x}$,½+z)	32	Pb.2	37	Ccc2
		($\overline{x}$, $\overline{y}$, z)	(½-x,½+y, z)	(½-y,½-x,½+z)	( $\overline{y}$, x,½+z)
107	I4mm	(x, y, z)	( x, $\overline{y}$, z)	( y, x, z)	( y, $\overline{x}$, z)	44	Imm2	42	Fmm2
		($\overline{x}$, $\overline{y}$, z))	( $\overline{x}$, y, z)	( $\overline{y}$, $\overline{x}$, z)	( $\overline{y}$, x, z)
108	14cm	(x, y, z)	(½+x,½-y, z)	(½+y,½+y, z)	( y, $\overline{x}$, z)	45	Iba2	42	Fmm2
		($\overline{x}$, $\overline{y}$, z)	(½-x,½+y, z)	(½-y,½-y, z)	( $\overline{y}$, x, z)
109	I41md	(x, y, z)	( x, $\overline{y}$, z)	( y,½+x,¼+z)	( y,½-x,¼+z)	44	Imm2	43	Fdd2
		($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y, z)	( $\overline{y}$, ½-x,¼+z)	( $\overline{y}$, ½+x,¼+z)
110	I41cd	(x, y, z)	( x, $\overline{y}$,½+z)	( y,½+x,⅔+z)	( y,½-x,¼+z)	45	Iba2	43	Fdd2
		($\overline{x}$, $\overline{y}$, z)	( $\overline{x}$, y,½+z)	( $\overline{y}$, ½-x,⅔+z)	( $\overline{y}$, ½+x,¼+z)
No.	Symbol	1st Subset	2d Subset	1st Subset	2d Subset			No,	Symbol
Space group of unstrained crystal, order 8 per lattice point		Coordinates referred to axes of unstrained crystal						Space group of strained crystal, order 4 per lattice point
		If stressed so that [l$\overline{1}$0], [110], z ∥ X,Y, Z, any permutation

Table 13.

Reduction from 422 to 222

Space group of unstrained crystal. order 8 per lattice point		If stressed so that x, y, z ∥ X, Y, Z, any permutation				Space group of strained crystal. order 4 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol		1st Subset		2d Subset	No.	Symbol
89	P422	(x,y, z) ($\overline{x}$, $\overline{y}$, z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	(y,x, $\overline{z}$) ($\overline{y}$,$\overline{x}$, $\overline{z}$)	( y, $\overline{x}$, z) ( $\overline{y}$, x, z)	16	P222	21	C222
90	P4212	(x,y, z) ($\overline{x}$$\overline{y}$, z)	(½+x, ½-y,$\overline{z}$) (½-x, ½+y,$\overline{z}$)	(y,x, $\overline{z}$) ($\overline{y}$,$\overline{x}$, $\overline{z}$)	(½+y, ½-x, z) (½-y, ½+x, z)	18	P21212	21	C222
91	P4122	(x,y, z) (x $\overline{y}$, ½+ z)	( x, $\overline{y}$, ½-z) ( $\overline{x}$, y, $\overline{z}$)	(y,x,¾- z) ($\overline{y}$,$\overline{x}$,¼- z)	( y, $\overline{x}$,¾+ z) ( $\overline{y}$, x,¼+ z)	17	P2221	20	C2221
92	P41212	(x,y, z) ($\overline{x}$$\overline{y}$, ½+ z)	(½+x, ½-y, ¾-z) (½-x, ½+y, ¼-z)	(y,x, $\overline{z}$) ($\overline{y}$,$\overline{x}$, ½-z)	(½+y, ½-x, ¾-z) (½-y, ½+x, ¼-z)	19	P212121	20	C2221
93	P4222	(x,y, z) ($\overline{x}$$\overline{y}$, z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	(y,x,½-z) ($\overline{y}$,$\overline{x}$,½-z)	( y, $\overline{x}$,½+z) ( $\overline{y}$, x,½+z)	16	P222	21	C222
94	P42212	(x,y, z) ($\overline{x}$,$\overline{y}$, z)	(½+x, ½-y, ½-z) (½-x, ½+y, ½-z)	(y,x, $\overline{z}$) ($\overline{y}$,$\overline{x}$, $\overline{z}$)	(½+y, ½-x, ½+z) (½-y, ½+x, ½+z)	18	P21212	21	C222
95	P4322	(x,y, z) ($\overline{x}$$\overline{y}$, ½+ z)	( x, $\overline{y}$, ½-z) ( $\overline{x}$, y, $\overline{z}$)	(y,x,¾- z) ($\overline{y}$,$\overline{x}$,¼- z)	( y, $\overline{x}$,¼+ z) ( $\overline{y}$, x,¾+ z)	17	P2221	20	C2221
96	P43212	(x,y, z) ($\overline{x}$$\overline{y}$, ½+ z)	(½+x, ½-y, ¼-z) (½-x, ½+y, ¾-z)	(y,x, $\overline{z}$) ($\overline{y}$,$\overline{x}$,½-z)	(½+y, ½-x, ¼-z) (½-y, ½+x, ¾-z)	19	P212121	20	C2221
97	1422	(x,y, z) ($\overline{x}$$\overline{y}$, z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	(y,x, $\overline{z}$) ($\overline{y}$,$\overline{x}$, $\overline{z}$)	( y, $\overline{x}$, z) ( $\overline{y}$, x, z)	23	I222	22	F222
98	14122	(x,y, z) ($\overline{x}$$\overline{y}$, z)	( x, ½-y, ¼-z) ( $\overline{x}$, ½+y, ¼-z)	(y,x, $\overline{z}$) ($\overline{y}$,$\overline{x}$, $\overline{z}$)	( y, ½+x, ¼+z) ( $\overline{y}$, ½+x, ¼+z)	24	I212121	22	F222
No.	Symbol	1st Subset	2d Subset	1st Subset	2d Subset			No.	Symbol
Space group of unstrained crystal, order 8 per lattice point		Coordinates referred to axes of unstrained crystal						Space group of strained crystal, order 4 per lattice point
		If stressed so that [1$\overline{1}$0], [110], z ∥ X, Y, Z, any permutation

Table 14.

Reduction from 6 to 2

Space group of unstrained crystal, order 6 per lattice point		If stressed so that z ∥ X, Y, or Z			Space group of strained crystal, order 2 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	No.	Symbol
168	P6	(x,y, z) ($\overline{x}$$\overline{y}$, z)	($\overline{y}$,x-y, z) (y,y-x, z)	(y-x,$\overline{x}$, z) (x-y,x, z)	3	P2
169	P61	(x,y, z) ($\overline{x}$$\overline{y}$, ½+ z)	($\overline{y}$,x-y, ⅓+ z) (y,y-x,⅚+ z)	(y-x,$\overline{x}$, ⅔+z) (x-y,x,⅙+ z)	4	P21
170	P65	(x,y, z) ($\overline{x}$$\overline{y}$, ½+ z)	($\overline{y}$,x-y,⅔+z) (y,y-x,⅙+ z)	(y-x,$\overline{x}$, ⅓+ z) (x-y,x,⅚+ z)	4	P21
171	P62	(x,y, z) ($\overline{x}$$\overline{y}$, z)	($\overline{y}$,x-y, ⅔+z) (y,y-x, ⅔+z)	(y-x,$\overline{x}$, ⅓+ z) (x-y,x, ⅓+ z)	3	P2
172	P64	(x,y, z) ($\overline{x}$$\overline{y}$, z)	($\overline{y}$,x-y, ⅓+ z) (y,y-x, ⅓+ z)	(y-x,$\overline{x}$, ⅔+z) (x-y,x, ⅔+z)	3	P2
173	P63	(x,y, z) ($\overline{x}$$\overline{y}$, ½+ z)	($\overline{y}$,x-y, z) (y,y-x, ½+ z)	(y-x,$\overline{x}$, z) (x-y,x, ½+ z)	4	P21

Table 15.

Reduction from $\overline{6}$to m

Space group of unstrained crystal, order 6 per lattice point		If stressed so that z ∥ X, Y, or Z			Space group of strained crystal, order 2 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	No.	Symbol
174	P$\overline{6}$	(x, y, z) (x, y, z)	($\overline{y}$, x−y, z) ($\overline{y}$, x−y, $\overline{z}$)	(y−x, $\overline{x}$, z) (y−x, $\overline{x}$, $\overline{z}$)	6	Pm

Table 16.

Reduction from 3m to m

Space group of unstrained crystal, order 6 per lattice point		If stressed so that	Coordinates referred to axes of unstrained crystal			Space group of strained crystal, order 2 per lattice point
No.	Symbol		1st Subset	2d Subset	3d Subset	No.	Symbol
156	P3m1	[110] ǁ X, Y, or Z	(x,y, z)	(y–x, $\overline{x}$, z)	( $\overline{y}$,x-y, z)	8	Cm
			(y,x, z)	( x,x–y, z)	(y-x, y, z)
157	P31m	[1$\overline{1}$0] ǁ X, Y, or Z	($\overline{y}$, $\overline{x}$, z)	(y–x, $\overline{x}$, z)	( $\overline{y}$,x-y, z)	8	Cm
			(x,y, z)	( $\overline{x}$,y-x, z)	(x-y, $\overline{y}$, z)
158	P3c1	[110] ǁ X, Y, or Z	(x, y, z)	(y–x, $\overline{x}$, z)	( $\overline{y}$,x-y, z)	9	Cc
			($\overline{y}$,$\overline{x}$,½+z)	( x,x-y, ½ +z)	(y-x, y, ½ +z)
159	P31c	[1$\overline{1}$0] ǁ X, Y, or Z	(x, y, z)	(y-x, $\overline{x}$, z)	( $\overline{y}$,x-y, z)	9	Cc
			(y,x, ½+z)	($\overline{x}$,y–x, ½+z)	(x-y, $\overline{y}$, ½ +z)
160	R3m hex. axes	[110] ǁ X, Y, or Z	(x,y, z)	(y–x, $\overline{x}$, z)	( $\overline{y}$,x-y, z)	8	Cm
			($\overline{y}$,$\overline{x}$, z)	( x,x-y, z)	(y-x, y, z)
161	R3c hex. axes	[110] ǁ X, Y, or Z	(x, y, z)	(y-x, $\overline{x}$, z)	( $\overline{y}$,x-y, z)	9	Cc
			($\overline{y}$,$\overline{x}$,½+z)	( x, x-y, ½+z)	(y-x, y, ½ +z)

Table 17.

Reduction from 32 to 2

Space group of unstrained crystal, order 6 per lattice point		If stressed so that	Coordinates referred to axes of unstrained crystal			Space group of strained crystal, order 2 per lattice point
No.	Symbol		1st Subset	2d Subset	3d Subset	No.	Symbol
149	P312	[1$\overline{1}$0] ǁ X, Y, or Z	(x,y, z)	($\overline{y}$,x-y, z)	(y—x, $\overline{x}$, z)	5	C2
			($\overline{y}$, $\overline{x}$, $\overline{z}$)	(y-x, y, $\overline{z}$)	( x,x-y, $\overline{z}$)
150	P321	[110] ǁ X, Y, or Z	(x, y, z)	($\overline{y}$,x-y, z)	(y—x, $\overline{x}$, z)	5	C2
			(y,x, $\overline{z}$)	(x-y, $\overline{y}$, z)	($\overline{x}$, y-x, z)
151	P3112	[1$\overline{1}$0] ǁ X, Y, or Z	(x.y, z)	($\overline{y}$,x-y, ⅓+z)	($\overline{y}$-x, $\overline{x}$, ⅔+z)	5	C2
			($\overline{y}$, $\overline{x}$, ⅔-z)	(y-x, y, ⅓-z)	( x, x—y, z)
152	P3121	[110] ǁ X, Y, or Z	(x, y, z)	($\overline{y}$,x-y,⅓+z)	(y-x, $\overline{x}$, ⅔+z)	5	C2
			(y,x, z)	(x-y, y, ⅔-z)	( x,y-x, ⅓-z)
153	P3212	[1$\overline{1}$0 ǁ X, Y, or Z	(x.y, z)	($\overline{y}$,x-y, ⅔+z)	(y-x, $\overline{x}$, ⅓+z)	5	C2
			($\overline{y}$, $\overline{x}$, ⅓-z)	(y-x, y, ⅔-z)	( x,x-y, z)
154	P3221	[110] ǁ X, Y, or Z	(x, y, z)	($\overline{y}$, x-y, ⅔+z)	(y-x, $\overline{x}$, ⅓+z)	5	C2
			(y,x, $\overline{z}$)	(x-y, $\overline{y}$, ⅓-z)	($\overline{x}$,y-x, ⅔-z)
155 hex.	R32 axes	[110] ǁ X, Y, or Z	(x, y, z)	($\overline{y}$,x-y, z)	(y-x, $\overline{x}$, z)	5	C2
			(y,x, z)	(x-y, y, $\overline{z}$)	($\overline{x}$, y—x, z)

Table 18.

Reduction from 4 to 2

Space group of unstrained crystal, order 4 per lattice point		If stressed so that z ∥ X, Y, or Z		Space group of strained crystal. order 2 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	No.	Symbol
75	P4	(x,y, z)	($\overline{y}$, x, z)	3	P2
		($\overline{x}$, $\overline{y}$, z)	(y, $\overline{x}$, z)
76	P41	(x, y, z)	($\overline{y}$, x, ¼+z)	4	P21
		($\overline{x}$,y, ½+z)	(y, $\overline{x}$, ¾+z)
77	P42	(x, y, z)	($\overline{y}$, x, ½+z)	3	P2
		(x, y, z)	(y, $\overline{x}$, ½+z)
78	P43	(x, y, z)	($\overline{y}$, x, ¾+z)	4	P21
		(x, $\overline{y}$, ½+z)	(y, $\overline{x}$, ¼+2)
79	14	(x, y, z)	($\overline{y}$, x, z)	5	C2
		($\overline{x}$, $\overline{y}$,, z)	(y, $\overline{x}$, z)
80	141	(x, y, z)	($\overline{y}$, ½+x, ¼+z)	5	C2
		($\overline{x}$, $\overline{y}$, z)	(y, ½-x, ¼+z)

Table 19.

Reduction from $\overline{4}$ to 2

Space group of unstrained crystal, order 4 per lattice point		If stressed so that z ∥ X, Y, or Z		Space group of strained crystal, order 2 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	No.	Symbol
81	P$\overline{4}$	(x, y, z)	($\overline{y}$,x, $\overline{z}$)	3	P2
		($\overline{x}$, $\overline{y}$, z)	(y, $\overline{x}$, $\overline{z}$)
82	I$\overline{4}$	(x, y, z)	($\overline{y}$,x, $\overline{z}$)	5	C2
		($\overline{x}$, $\overline{y}$, z)	(y, $\overline{x}$, $\overline{z}$)

Table 20.

Reduction from mm2 to m

Space group of unstrained crystal, order 4 per lattice point		If stressed so that x ∥ X, Y, or Z		Space group of strained crystal, order 2 per lattice point		If stressed so that y ∥ X, Y, or Z		Space group of strained crystal, order 2 per lattice point
		Coordinates referred to axes of unstrained crystal				Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	No.	Symbol	1st Subset	2d Subset	No.	Symbol
25	Pmm2	( x, y, z)	( x, $\overline{y}$, z)	6	Pm	( x, y, z)	( x, y, z)	6	Pm
		( $\overline{x}$, y, z)	( $\overline{x}$, $\overline{y}$, z)			( x, y, z)	( x, y, z)
26	Pmc21	( x, y, z)	( x, $\overline{y}$, ½+z)	6	Pm	x, y, z)	( $\overline{x}$, y, z)	7	Pc
		( $\overline{x}$, y, z)	( x, $\overline{y}$, ½+z)			( x, $\overline{y}$, ½+z)	( $\overline{x}$, $\overline{y}$, ½+z)
27	Pcc2	( x, y, z)	( x, $\overline{y}$, ½+z)	7	Pc	( x, y, z)	( $\overline{x}$, y, ½+z)	7	Pc
		( $\overline{x}$, y, ½+z)	( $\overline{x}$, $\overline{y}$, z)			( x, $\overline{y}$, ½+z)	( $\overline{x}$, $\overline{y}$, z)
28	Pma2	( x, y, z)	(½+x, $\overline{y}$, z)	6	Pm	( x, y, z)	(½-x, y, z)	7	Pc
		(½-x, y, z)	( $\overline{x}$, $\overline{y}$, z)			(½+x, $\overline{y}$, z)	( $\overline{x}$, $\overline{y}$, z)
29	Pca21	( x, y, z)	(½+x, $\overline{y}$, z)	7	Pc	( x, y, z)	(½-x, y, ½+z)	7	Pc
		(½-x, y, ½+z)	( $\overline{x}$, $\overline{y}$, ½+z)			(½+x, $\overline{y}$, z)	( $\overline{x}$, $\overline{y}$, ½+z)
30	Pnc2	( x, y, z)	($\overline{x}$, ½-y, ½+z)	7	Pc	( x, y, z)	( $\overline{x}$, ½+y, ½+z)	7	Pc
		( $\overline{x}$, ½+y, ½+z)	( $\overline{x}$, $\overline{y}$, z)			( x, ½-y, ½+z)	( $\overline{x}$, $\overline{y}$, z)
31	Pmn21	( x, y, z)	(½+x, $\overline{y}$, ½+z)	6	Pm	( x, y, z)	( $\overline{x}$, y, z)	7	Pc
		( x, y, z)	(½-x, $\overline{y}$, ½+z)			(½+x, $\overline{y}$, ½+z)	(½-x, $\overline{y}$, ½+z)
32	Pba2	( x, y, z)	(½+x, ½-y, Z)	7	Pc	( x, y, z)	(½-x, ½+y, z)	7	Pc
		(½-x, ½+y, z)	( $\overline{x}$, $\overline{y}$, z)			(½+x, ½-y, z)	( $\overline{x}$, $\overline{y}$, z)
33	Pna21	( x, y, z)	(½+x, ½-y, z)	7	Pc	( x, y, z)	(½-x, ½+y, ½+z)	7	Pc
		(½-x, ½+y, ½+z)	( $\overline{x}$, $\overline{y}$, ½+z)			(½+x, ½-y, Z)	( $\overline{x}$, $\overline{y}$, ½+z)
34	Pnn2	( X, y, z)	(½+x, ½-y, ½+z)	7	Pc	( X, y, z)	(½-x, ½+y, ½+z)	7	Pc
		(½-x, ½+y,½+z)	( $\overline{x}$, $\overline{y}$, z)			(½+x, ½-y, ½+z)	( $\overline{x}$, $\overline{y}$, z)
35	Cmm2	( X, y, z)	( x, $\overline{y}$, z)	8	Cm	( x, y, z)	( $\overline{x}$, y, z)	8	Cm
		( $\overline{x}$, y, z)	( $\overline{x}$, $\overline{y}$, z)			( x, $\overline{y}$, z)	( $\overline{x}$, $\overline{y}$, z)
36	Cmc21	( x, y, z)	( x, $\overline{y}$, ½+z)	8	Cm	( x, y, z)	( $\overline{x}$, y, z)	9	Cc
		( $\overline{x}$, y, z)	( $\overline{x}$, $\overline{y}$, ½+z)			( x, $\overline{y}$, ½+z)	( $\overline{x}$, $\overline{y}$, ½+z)
37	Ccc2	( x, y, z)	( x, $\overline{y}$, ½+z)	9	Cc	( x, y, z)	( $\overline{x}$, y, ½+z)	9	Cc
		( $\overline{x}$, y, ½+z)	( $\overline{x}$, $\overline{y}$, z)			( x, $\overline{y}$, ½+z)	( $\overline{x}$, $\overline{y}$, z)
38	Amm2	( x, y, z)	( x, $\overline{y}$, z)	6	Pm	( x, y, z)	( $\overline{x}$, y, z)	8	Cm
		( $\overline{x}$, y, z)	( $\overline{x}$, $\overline{y}$, z)			( x, $\overline{y}$, z)	( $\overline{x}$, $\overline{y}$, z)
39	Abm2	( x, y, z)	( x, ½-y, z)	7	Pc	( x, y, z)	( $\overline{x}$, ½+y, z)	8	Cm
		( $\overline{x}$, ½+y, z)	( $\overline{x}$, $\overline{y}$, z)			( x, ½-y, z)	( $\overline{x}$, $\overline{y}$, z)
40	Ama2	( x, y, z)	(½+x, $\overline{y}$, z)	6	Pm	( x, y, z)	(½-x, y, z)	9	Cc
		(½-x, y, z)	( $\overline{x}$, $\overline{y}$, z)			(½+x, $\overline{y}$, z)	( $\overline{x}$, $\overline{y}$, z)
41	Aba2	( x, y, z)	(½+x, ½-y, z)	7	Pc	( x, y, z)	(½-x, ½+y, z)	9	Cc
		(½-x, -+y, z)	( $\overline{x}$, $\overline{y}$, z)			(½+x, ½-y, z)	( $\overline{x}$, $\overline{y}$, z)
42	Fmm2	( x, y, z)	( x, $\overline{y}$, z)	8	Cm	( x, y, z)	( $\overline{x}$, y, z)	8	Cm
		( $\overline{x}$, y, z)	( $\overline{x}$, $\overline{y}$, z)			( x, $\overline{y}$, z)	( $\overline{x}$, $\overline{y}$, z)
43	Fdd2	( x, y, z)	(¼+x, ¼-y, ¼+z)	9	Cc	( x, y, z)	(¼-x, ¼+y, ¼+z)	9	Cc
		(¼-x, ¼+y, ¼+z)	( $\overline{x}$, $\overline{y}$, z)			(¼+x, ¼-y, ¼+z)	( $\overline{x}$, $\overline{y}$, z)
44	Imm2	( x, y, z)	( x, $\overline{y}$, z)	8	Cm	( x, y, z)	( $\overline{x}$, y, z)	8	Cm
		( $\overline{x}$, y, z)	( $\overline{x}$, $\overline{y}$, z)			( x, $\overline{y}$, z)	( $\overline{x}$, $\overline{y}$, z)
45	Iba2	( x, y, z)	( x, $\overline{y}$, ½+z)	9	Cc	( x, y, z)	( $\overline{x}$, y, ½+z)	9	Cc
		( $\overline{x}$, y, ½+z)	( $\overline{x}$, $\overline{y}$, z)			( x, $\overline{y}$, ½+z)	( $\overline{x}$, $\overline{y}$, z)
46	Ima2	( x, y, z)	(½+x, $\overline{y}$, z)	8	Cm	( x, y, z)	(½-x, y, z)	9	Cc
		( ½−x, y, z)	( $\overline{x}$, $\overline{y}$, z)			( ½+x, $\overline{y}$, z)	( $\overline{x}$, $\overline{y}$, z)

Table 21.

Reduction from mm2 to 2

Space group of unstrained crystal, order 4 per lattice point		If ,tressed so that z ∥ X. Y, or Z		Space group of strained crystal, order 2 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	No.	Symbol
25	Pmm2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, z) ( $\overline{x}$, y, z)	3	P2
26	Pmc21	( x,y, z) ( $\overline{x}$,$\overline{y}$, ½+z)	( x, $\overline{y}$, ½+z) ( $\overline{x}$, y, z)	4	P21
27	Pcc2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, ½+z) ( $\overline{x}$, y, ½+z)	3	P2
28	Pma2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	(½+x, $\overline{y}$, z) (½-x, y, z)	3	P2
29	Pca21	( x,y, z) ( $\overline{x}$,$\overline{y}$, ½+z)	(½+x, $\overline{y}$, z) (½-x, y, ½+z)	4	P21
30	Pnc2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, ½ -y, ½+z) ($\overline{x}$, ½+-y, ½+z)	3	P2
31	Pmn21	( x,y, z) ( x,y, ½+z)	(½+x, y, ½+z) ( $\overline{x}$, y, z)	4	P21
32	Pba2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	(½+x,½-y, z) (½-x,½+y, z)	3	P2
33	Pna21	( x,y, z) ( $\overline{x}$,$\overline{y}$, ½+z)	(½+x,½-y, z) (½-x,½+y, ½+ z)	4	P21
34	Pnn2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	(½+x,½-y, ½+ z) (½-x,½+y, ½+ z)	3	P2
35	Cmm2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, z) ( $\overline{x}$, y, z)	3	P2
36	Cmc21	( x,y, z) ( $\overline{x}$,$\overline{y}$, ½+z)	( x, $\overline{y}$, ½+z) ( $\overline{x}$, y, z)	4	P21
37	Ccc2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, ½+z) ( $\overline{x}$, y, ½+z)	3	P2
38	Amm2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, z) ( $\overline{x}$, y, z)	5	C2
39	Abm2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, ½-y, z) ( $\overline{x}$, ½+y, z)	5	C2
40	Ama2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	(½+x, $\overline{y}$, z) (½-x, y, z)	5	C2
41	Aba2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	(½+x,½-y, z) (½-x,½+y, z)	5	C2
42	Fmm2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, z) ( $\overline{x}$, y, z)	5	C2
43	Fdd2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	(¼+x, ¼-y, ¼+ z) (¼-x, ¼+y, ¼+ z)	5	C2
44	Imm2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, z) ( $\overline{x}$, y, z)	5	C2
45	Iba2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, ½+z) ( $\overline{x}$, y, ½+z)	5	C2
46	Ima2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	(½+x, $\overline{y}$, z) (½-x, y, z)	5	C2

Table 22.

Reduction from 222 to 2

Space group of unstrained crystal, order 4 per lattice point		If stressed so that x ∥ X, Y, or Z		Space group of strained crystal, order 2 per lattice point		If stressed so that y ∥ X, Y, or Z		Space group of strained crystal, order 2 per lattice point		If stressed so that z ∥ X, Y, or Z		Space group of strained crystal, order 2 per lattice point
		Coordinates referred to axes of unstrained crystal				Coordinates referred to axes of unstrained crystal				Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	No.	Symbol	1st Subset	2d Subset	No.	Symbol	1st Subset	2d Subset	No.	Symbol
16	P222	( x, y, z) ( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	3	P2	( x, y, z) ( $\overline{x}$, y, $\overline{z}$)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	3	P2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	3	P2
17	P2221	( x, y, z) ( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, ½- z) ( $\overline{x}$, $\overline{y}$, ½+ z)	3	P2	( x, y, z) ( $\overline{x}$, y, ½- z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, ½+ z)	3	P2	( x,y, z) ( $\overline{x}$,$\overline{y}$, ½+ z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, y, ½- z)	4	P21
18	P21212	( x, y, z) (½+x, ½-y, $\overline{z}$)	(½-x, ½+ y, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	4	P21	( x, y, z) (½-x, ½+y, $\overline{z}$)	(½+x, ½- y, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	4	P21	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	(½+x, ½- y, $\overline{z}$) (½-x, ½+ y, $\overline{z}$)	3	P2
19	P212121	( x, y, z) (½+x, ½-y, $\overline{z}$)	( $\overline{x}$, ½+ y, ½- z) (½-x, $\overline{y}$, ½+ z)	4	P21	( x, y, z) ( $\overline{x}$, ½+y , ½- z)	(½+x, ½- y, $\overline{z}$) (½-x, $\overline{y}$, ½+ z)	4	P21	( x,y, z) (½-x,$\overline{y}$, ½+z)	(½+x, ½- y, $\overline{z}$) ( $\overline{x}$,½+y, ½- z)	4	P21
20	C2221	( x, y, z) ( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, ½- z) ( $\overline{x}$, $\overline{y}$, ½+ z)	5	C2	( x, y, z) ( $\overline{x}$, y, ½- z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, ½+ z)	5	C2	( x,y, z) ( $\overline{x}$,$\overline{y}$, ½+ z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, y, ½- z)	4	P21
21	C222	( x, y, z) ( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	5	C2	( x, y, z) ( $\overline{x}$, y, $\overline{z}$)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	5	C2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	3	P2
22	F222	( x, y, z) ( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	5	C2	( x, y, z) ( $\overline{x}$, y, $\overline{z}$)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	5	C2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	5	C2
23	I222	( x, y, z) ( x, $\overline{y}$, $\overline{z}$)	( $\overline{x}$, y, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	5	C2	( x, y, z) ( $\overline{x}$, y, $\overline{z}$)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, $\overline{y}$, z)	5	C2	( x,y, z) ( $\overline{x}$,$\overline{y}$, z)	( x, $\overline{y}$, $\overline{z}$) ( $\overline{x}$, y, $\overline{z}$)	5	C2
24	I212121	( x, y, z) (½+x, ½-y, $\overline{z}$)	( $\overline{x}$, ½+ y, ½- z) (½-x, $\overline{y}$, ½+ z)	5	C2	( x, y, z) ( $\overline{x}$, ½+y , ½+ z)	(½+x, ½- y, $\overline{z}$) (½-x, $\overline{y}$, ½+ z)	5	C2	( x,y, z) (½-x,$\overline{y}$, ½+z)	(½+x, ½- y, $\overline{z}$) ( $\overline{x}$,½+y, ½- z)	5	C2

Table 23.

Reduction from 3 to 1

Space group of unstrained crystal, order 3 per lattice point		No specialization of stress			Space group of strained crystal, order 1 per lattice point
		Coordinates referred to axes of unstrained crystal
No.	Symbol	1st Subset	2d Subset	3d Subset	No.	Symbol
143	P3	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	1	P1
144	P31	(x, y, z)	($\overline{y}$,x-y,⅓+ z)	(y-x,$\overline{x}$, ⅔+z)	1	P1
145	P31	(x, y, z)	($\overline{y}$,x-y, ⅔+z)	(y-x,$\overline{x}$, ⅓+z)	1	P1
146	R3 hex. axes	(x, y, z)	($\overline{y}$,x-y, z)	(y-x,$\overline{x}$, z)	1	P1

Table 24.

Reduction from m to 1

Space group of unstrained crystal, order 2 per lattice point		No specialization of stress		Space group of strained crystal, order 1 per lattice point
		Coordinates referred to axes of unstrained crystal (m⊥y)
No.	Symbol	1st Subset	2d Subset	No.	Symbol
6	Pm	(x, y, z)	(x, $\overline{y}$, z)	1	P1
7	Pc	(x, y, z)	(x, $\overline{y}$, ½+ z)	1	P1
8	Cm	(x, y, z)	(x, $\overline{y}$, z)	1	P1
9	Cc	(x, y, z)	(x, $\overline{y}$, ½+ z)	1	P1

Table 25.

Reduction from 2 to 1

Space group of unstrained crystal, order 2 per lattice point		No specialization of stress		Space group of strained crystal, order 1 per lattice point
		Coordinates referred to axes of unstrained crystal (2∥Y)
No.	Symbol	1st Subset	2d Subset	No.	Symbol
3	P2	(x, y, z)	( $\overline{x}$, y, $\overline{z}$ )	1	P1
4	P21	(x, y, z)	( $\overline{x}$x,½+ y, $\overline{z}$ )	1	P1
5	C2	(x, y, z)	( $\overline{x}$, y, $\overline{z}$ )	1	P1

2.2. Stress Table

The most general stress consistent with each possible step of stress-induced symmetry lowering, minimum or compound, is listed in table 26 for all of of the noncentrosymmetric point groups. The stress conditions for the minimum steps of stress-induced symmetry lowering are equivalent, though not always identical, to those given in tables 1–25.

Table 26.

Most general stress consistent with a reduction of a noncentrosymmetric point group to any one of its stress-induced subgroups

Large square brackets indicate same subgroup in equivalent setting.