International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.6, pp. 854856
Section 3.6.2.2.4. Opechowski–Guccione magnetic group type symbols and the standard set of coset representatives^{a}Department of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA 
3.6.2.2.4. Opechowski–Guccione magnetic group type symbols and the standard set of coset representatives
The specification of the magnetic group type symbol and the standard set of coset representatives of the magnetic group type's representative group is based on the conventions introduced by Opechowski and Guccione (Opechowski & Guccione, 1965; Opechowski, 1986) for threedimensional magnetic space groups. The specification was made in conjunction with Volume I of International Tables for Xray Crystallography (1969) (abbreviated here as ITXC I). One finds in ITXC I, for each group type , a specification of the coordinate system used, and, in terms of that coordinate system, a specification of the subgroup of translations of the representative space group of that group type, and also indirectly a specification of a set of coset representatives of of that representative group of group type . These coset representatives are uniquely determined from the coordinate triplets of the explicitly printed general position of the space group. The symbol for the space group is taken to be the spacegroup symbol at the top of the page listing these coordinate triplets. The symbol for a group type is that of the group type followed by , and the coset representatives of the representative group of the group type consist of the set of coset representatives of and this set multiplied by .
Example
In ITXC I, on the page for one finds the following coordinate triplets of the general position:determining the coset representative of the representative group :The coset representatives of the representative group are then:
ITXC I has been replaced by IT A. Ones finds that, for some space groups, the set of coordinate triplets of the general positions explicitly printed in IT A differs from that explicitly printed in ITXC I. As a consequence, if one attempts to interpret the Opechowski–Guccione symbols (OG symbols) for magnetic groups using IT A, one will, in many cases misinterpret the meaning of the symbol (Litvin, 1997, 1998). [It was suggested in these two papers that the original set of OG symbols should be modified so one could correctly interpret them using IT A instead of ITXC I. Adopting this illadvised suggestion would have required in the future a new modification of the OG symbols whenever changes were made to the choices of coordinate triplets of the general position in IT A. Consequently, the meaning of the original OG symbols was specified by Litvin (2001) by explicitly giving the coset representatives of the representative groups of each threedimensional magnetic space group.]
The symbol for a magnetic group type and its representative group is based on the symbol for the group type . is an equitranslational subgroup of , i.e. the translational subgroup of the magnetic group is , the translational subgroup of . The translational part of the group type symbol of an group is then the same as that of the group type . A number or letter in the remaining part of the symbol of appears unchanged in the symbol for if it is associated with a coset representative of the representative group that is also an element contained in the subgroup of . If not in , i.e. if in , the number or letter appears in the symbol for with a prime to denote that the element in is coupled with .
Example
The orthorhombic spacegroup type has the magnetic spacegroup type number 29.1.198. The representative group is defined by a orthorhombic translational subgroup denoted by the letter P in Pca2_{1} and the standard set of coset representativesThe magnetic spacegroup type 29.5.202 is a group whose symbol is . In this case we have , i.e. and . The symbol `2_{1}' in the symbol for refers to the coset representative {2_{001}∣0, 0, ½}, an element in . Consequently, the symbol `2_{1}' appears unprimed in the symbol for () and the coset representative {2_{001}∣0, 0, ½} appears as an unprimed coset representative in the standard set of coset representatives of . The symbols `c' and `a' in refer to the coset representatives {m_{100}∣½, 0, ½} and {m_{010}∣½, 0, 0}, respectively, neither of which are contained in . Consequently, both symbols appear primed in the symbol for and the coset representatives {m_{100}∣½, 0, ½} and {m_{010}∣½, 0, 0} appear as primed coset representatives in the standard set of coset representatives of . The representative magnetic space group then has the orthorhombic translational subgroup denoted by the letter P and the standard set of coset representatives
The symbol for a group type and its representative group is also based on the symbol for the group . [This is in contradistinction to the BNS symbols of groups (Belov et al., 1957), where the symbol for an group type is based on the symbol for the group , see Section 3.6.4.] As this is an equiclass magnetic group, half the translations of are now coupled with in and half the translations remain unprimed in . The unprimed translations constitute the translational subgroup of . We can then write the coset decomposition of the translational subgroup of with respect to the translational subgroup of aswhere denotes a chosen coset representative, a translation of which appears primed (coupled with ) in . The translational subgroup of can then be written asSymbols for the translational groups , the translational subgroups of , the translational groups of and the choice of the translations are given in Fig. 1 of Litvin (2013).
The symbol for a magnetic group type and its representative group is based on the symbol of the group type , and is also a symbol for the subgroup of unprimed elements: the translational part of the symbol of is replaced by the symbol for the translational subgroup of . If a coset representative {R∣τ(R)} of in appears as the coset representative {R∣τ(R) + t_{α}} of in , then the number or letter corresponding to {R∣τ(R)} in the symbol for is primed. If {R∣τ(R)} appears unchanged as a coset representative of in , then the number or letter corresponding to {R∣τ(R)} in the symbol for is unchanged (Opechowski & Litvin, 1977). The resulting symbol is a symbol for based on the symbol for and is also a symbol for the magnetic group : the symbol specifies not only but also ; by deleting the subindex on the translational part of the symbol and the primes on the rotational part, one obtains the symbol specifying . Having specified and , one has specified the group .
Example
Consider again the group 29.1.198, , whereThe symbol for the group type 29.7.204 is and is based on the symbol for . The translational subgroup of is given by the symbol P_{2b} where = {1∣0, 1, 0}, i.e. is generated by the three translations {1∣1, 0, 0}, {1∣0, 2, 0} and {1∣0, 0, 1} of , and the translational subgroup of is given by . The two primed symbols and in refer to the fact that the two coset representatives {m_{100}∣½, 0, ½} and {m_{010}∣½, 0, 0} that appear in the set of standard coset representatives of in appear as the coset representatives {m_{100}∣½, 1, ½} and {m_{010}∣½, 1, 0} in the set of standard coset representatives of in . As the symbol 2_{1} in is not primed, the coset representative {2_{001}∣0, 0, ½} of in remains unchanged as a coset representative of in . We have then the subgroupWe note that these same coset representatives of in are also the coset representatives of the standard set of coset representatives of in .and the standard set of coset representatives of listed in the tables is the same set of coset representatives:Since , it follows that andConsequently, a primed number or letter in the symbol for (which is also a symbol for ) denotes that the corresponding coset representative appears in coupled with and primed in , e.g. in denotes that the coset {m_{100}∣½, 0, ½} appears as {m_{100}∣½, 1, ½} in and as {m_{100}∣½, 0, ½}′ in . An unprimed number or letter in the symbol for (which is also a symbol for ) denotes that the corresponding element appears unchanged in and coupled with and primed in , the symbol 2_{1} in denotes that {2_{001}∣0, 0, ½} is in and {2_{001}∣1, 0, ½}′ is in .
For twodimensional magnetic space groups with square and hexagonal lattices, threedimensional magnetic space groups with tetragonal, hexagonal, rhombohedral and cubic lattices, and magnetic layer and rod groups with tetragonal, trigonal or hexagonal lattices, each letter or number in the group type symbol may represent not a single symmetry direction but a set of symmetry directions, see Table 1.3 in Litvin (2013), Table 2.1.3.1 in the present volume and Table 1.2.4.1 in IT E. Stokes & Campbell (2009) have pointed out that this can lead to not being able to determine the standard set of coset representatives in groups from the Opechowski–Guccione magnetic group symbol. Consequently, we introduce the convention that in determining the standard set of coset representatives, each letter or number in the group type symbol in these groups refers to the first symmetry direction of each set of symmetry directions listed in the aforementioned tables.
Example
The standard set of coset representatives of 94.1.786 P4_{2}2_{1}2 isFor the threedimensional magnetic group 94.7.792 , the standard set of coset representatives isThe secondary position in the Hermann–Mauguin symbol denotes the set of symmetry directions {[100], [010]}. With this convention, the primed symbol denotes that the corresponding coset representative {2_{100}∣½, ½, ½} of P4_{2}2_{1}2 appears in the coset representatives of coupled with the translation = {1∣0, 0, 1}. The third position in the Hermann–Mauguin symbol denotes the set of symmetry directions . The unprimed symbol 2 denotes that the coset representative of P4_{2}2_{1}2 appears unchanged in the coset representatives of .
The impact of this convention is that nine Opechowski–Guccione symbols of threedimensional magnetic space groups need to be changed:

To have all group symbols represent subgroups , six symbols for threedimensional magnetic space groups were based (Opechowski & Guccione, 1965) on the symbol of the subgroup instead of the symbol for . These are the groups 144.3.1236 P_{2c}3_{2}, 145.3.1239 P_{2c}3_{1}, 151.4.1262 P_{2c}3_{2}12, 152.4.1266 P_{2c}3_{2}21, 153.4.1270 P_{2c}3_{1}12 and 154.4.1274 P_{2c}3_{1}21. Additional groups are the rod groups 43.3.231 , 44.3.234 , 47.4.246 and 48.4.250 .
References
International Tables for Xray Crystallography (1969). Vol. I, 3rd ed., edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Earlier editions 1952, 1965.]Belov, N. V., Neronova, N. N. & Smirnova, T. S. (1957). 1651 Shubnikov groups. Sov. Phys. Crystallogr. 1, 487–488. See also (1955) Trudy Inst. Krist. Acad. SSSR, 11, 33–67 (in Russian), English translation in Shubnikov, A. V, Belov, N. V. & others (1964). Colored Symmetry. London: Pergamon Press.
Litvin, D. B. (1997). Changes in the Opechowski–Guccione symbols for magnetic space groups due to changes in the International Tables of Crystallography. Ferroelectrics, 204, 211–215.
Litvin, D. B. (1998). On Opechowski–Guccione magnetic spacegroup symbols. Acta Cryst. A54, 257–261.
Litvin, D. B. (2001). Magnetic spacegroup types. Acta Cryst. A57, 729–730.
Litvin, D. B. (2013). Magnetic Group Tables, 1, 2 and 3Dimensional Magnetic Subperiodic Groups and Space Groups. Chester: International Union of Crystallography. Freely available from http://www.iucr.org/publ/978–0–9553602–2–0 .
Opechowski, W. (1986). Crystallographic and Metacrystallographic Groups. Amsterdam: North Holland.
Opechowski, W. & Guccione, R. (1965). Magnetic symmetry. In Magnetism, edited by G. T. Rado & H. Suhl, Vol. 2A, ch. 3. New York: Academic Press.
Opechowski, W. & Litvin, D. B. (1977). Error corrections corrected: A remark on Bertaut's article, Simple derivation of magnetic space groups. Ann. Phys. 2, 121–125.
Stokes, H. T. & Campbell, B. J. (2009). Table of magnetic space groups, http://stokes.byu.edu/magneticspacegroupshelp.html.