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. 852865
https://doi.org/10.1107/97809553602060000934 Chapter 3.6. Magnetic subperiodic groups and magnetic space groups^{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 An introduction to magnetic groups and to tables of these magnetic groups is presented. The magnetic groups considered are the magnetic point groups, the two and threedimensional magnetic subperiodic groups, i.e. the magnetic frieze, rod and layer groups, and the one, two and threedimensional magnetic space groups. The structure, symbols and properties of these magnetic groups and the maximal subgroups listed in the tables are defined, and the Opechowski–Guccione and Belov–Nerenova–Smirnova magnetic group symbols are compared. 
The magnetic subperiodic groups in the title refer to generalizations of the crystallographic subperiodic groups, i.e. frieze groups (twodimensional groups with onedimensional translations), crystallographic rod groups (threedimensional groups with onedimensional translations) and layer groups (threedimensional groups with twodimensional translations). There are seven friezegroup types, 75 rodgroup types and 80 layergroup types, see International Tables for Crystallography, Volume E, Subperiodic Groups (2010; abbreviated as IT E). The magnetic space groups refer to generalizations of the one, two and threedimensional crystallographic space groups, ndimensional groups with ndimensional translations. There are two onedimensional spacegroup types, 17 twodimensional spacegroup types and 230 threedimensional spacegroup types, see Part 2 of the present volume (IT A).
Generalizations of the crystallographic groups began with the introduction of an operation of `change in colour' and the `twocolour' (black and white, antisymmetry) crystallographic point groups (Heesch, 1930; Shubnikov, 1945; Shubnikov et al., 1964). Subperiodic groups and space groups were also extended into twocolour groups. Twocolour subperiodic groups consist of 31 twocolour friezegroup types (Belov, 1956a,b), 394 twocolour rodgroup types (Shubnikov, 1959a,b; Neronova & Belov, 1961a,b; Galyarski & Zamorzaev, 1965a,b) and 528 twocolour layergroup types (Neronova & Belov, 1961a,b; Palistrant & Zamorzaev, 1964a,b). Of the twocolour space groups, there are seven twocolour onedimensional spacegroup types (Neronova & Belov, 1961a,b), 80 twocolour twodimensional spacegroup types (Heesch, 1929; Cochran, 1952) and 1651 twocolour threedimensional spacegroup types (Zamorzaev, 1953, 1957a,b; Belov et al., 1957). See also Zamorzaev (1976), Shubnikov & Koptsik (1974), Koptsik (1966, 1967), and Zamorzaev & Palistrant (1980). [Extensive listings of references on colour symmetry, magnetic symmetry and related topics can be found in the books by Shubnikov et al. (1964), Shubnikov & Koptsik (1974), and Opechowski (1986).]
The socalled magnetic groups, groups to describe the symmetry of spin arrangements, were introduced by Landau & Lifschitz (1951, 1957) by reinterpreting the operation of `change in colour' in twocolour crystallographic groups as `time inversion'. This chapter introduces the structure, properties and symbols of magnetic subperiodic groups and magnetic space groups as given in the extensive tables by Litvin (2013), which are an extension of the classic tables of properties of the two and threedimensional subperiodic groups found in IT E and the one, two and threedimensional space groups found in the present volume. A survey of magnetic group types is also presented in Litvin (2013), listing the elements of one representative group in each reduced superfamily of the two and threedimensional magnetic subperiodic groups and one, two and threedimensional magnetic space groups. Two notations for magnetic groups, the Opechowski–Guccione notation (OG notation) (Guccione, 1963a,b; Opechowski & Guccione, 1965; Opechowski, 1986) and the Belov–Neronova–Smirnova notation (BNS notation) (Belov et al., 1957) are compared. The maximal subgroups of index 4 of the magnetic subperiodic groups and magnetic space groups are also given.
We review the concept of a reduced magnetic superfamily (Opechowski, 1986) to provide a classification scheme for magnetic groups. This is used to obtain the survey of the two and threedimensional magnetic subperiodic group types and the one, two and threedimensional magnetic space groups given in Litvin (2013). In that survey a specification of a single representative group from each group type is provided.
Let denote a crystallographic group. The magnetic superfamily of consists of the following set of groups:
The third subset is divided into two subdivisions:

Two magnetic groups and are called equivalent if there exists an affine transformation that maps onto and onto (Opechowski, 1986). If only nonequivalent groups are included, then the above set of groups is referred to as the reduced magnetic superfamily of .
Example
We consider the crystallographic point group . The magnetic superfamily of the group consists of five groups: , the group , and the three groups , and . Since the latter three groups are all equivalent, the reduced magnetic superfamily of the group consists of only three groups: , , and one of the three groups , and .
Example
In the reduced magnetic space group superfamily of there are five groups: , , and three groups , and . The groups and are equitranslational magnetic space groups and is an equiclass magnetic space group .
A magnetic group has been defined as a symmetry group of a spin arrangement (Opechowski, 1986). With this definition, since , a group is then not a magnetic group. However, there is not universal agreement on the definition or usage of the term magnetic group. Two definitions (Opechowski, 1986) have magnetic groups as symmetry groups of spin arrangements, with one having only groups , of the three types of groups , and , defined as magnetic groups, while a second having both group and defined as magnetic groups. Here we shall refer to all groups in a magnetic superfamily of a group as magnetic groups, while cognizant of the fact that groups cannot be a symmetry group of a spin arrangement.
The survey consists of listing the reduced magnetic superfamily of one group from each type of one, two and threedimensional crystallographic point groups, two and threedimensional crystallographic subperiodic groups, and one, two and threedimensional space groups (Litvin, 1999, 2001, 2013). The numbers of types of groups , and in the reduced superfamilies of these groups is given in Table 3.6.2.1. The one group from each type, called the representative group of that type, is specified by giving a symbol for its translational subgroup and listing a set of coset representatives, called the standard set of coset representatives, of the decomposition of the group with respect to its translational subgroup. The survey provides the following information for each magnetic group type and its associated representative group:

Examples of entries in the survey of magnetic groups are given in Table 3.6.2.2. The survey of the threedimensional magnetic space groups (Litvin, 2001, 2013) was incorporated into the survey of threedimensional magnetic space groups given by Stokes & Campbell (2009) and the coset representatives can also be found on the Bilbao Crystallographic Server (http://www.cryst.ehu.es ; Aroyo et al., 2006).

For each set of magnetic group types, one, two and threedimensional crystallographic magnetic point groups, magnetic subperiodic groups and magnetic space groups, a separate numbering system is used. A threepart composite number N_{1}.N_{2}.N_{3} is given in the first column, see Table 3.6.2.2. N_{1} is a sequential number for the group type to which belongs. N_{2} is a sequential numbering of the magnetic group types of the reduced magnetic superfamily of . Group types always have the assigned number N_{1}.1.N_{3}, and group types the assigned number N_{1}.2.N_{3}. N_{3} is a global sequential numbering for each set of magnetic group types. The sequential numbering N_{1} for subperiodic groups and space groups follows the numbering in IT E and IT A, respectively.
A Hermann–Mauguinlike type symbol is given for each magnetic group type in the second column. This symbol denotes both the group type and the representative group of that type. For example, the symbol for the threedimensional magnetic spacegroup type 25.4.158 is . This symbol denotes both the group type, which consists of an infinite set of groups, and the representative group . While this representative group may be referred to as `the group ', other groups of this group type, e.g. , will always be written with subindices. The representative group of the magnetic group type is defined by its translational subgroup, implied by the first letter in the magnetic group type symbol and defined in Table 1.1 of Litvin (2013), and a given set of coset representatives, called the standard set of coset representatives, of the representative group with respect to its translational subgroup.
Only the relative lengths and mutual orientations of the translation vectors of the translational subgroup are given, see Table 1.2 of Litvin (2013). The symmetry directions of symmetry operations represented by characters in the Hermann–Mauguin symbols are implied by the character's position in the symbol and are given in Table 1.3 of Litvin (2013). The standard set of coset representatives are given with respect to an implied coordinate system. The absolute lengths of translation vectors, the position in space of the origin of the coordinate system and the orientation in that space of the basis vectors of that coordinate system are not explicitly given.
The standard set of coset representatives of each representative group is listed on the righthand side of the survey of magnetic group types, see e.g. Table 3.6.2.2. Each coset in the standard set of coset representatives is given in Seitz notation (Seitz, 1934, 1935a,b, 1936), i.e. {R∣τ} or {R∣τ}′. R denotes a proper or improper rotation (rotationinversion), τ a nonprimitive translation with respect to the nonprimed translational subgroup of the magnetic group, and the prime denotes that {R∣τ} is coupled with time inversion. The subindex notation on R, denoting the orientation of the proper or improper rotation, is given in Table 1.4 of Litvin (2013). [Note that the Seitz notation used in Litvin (2013) predates and is different from the IUCr standard convention for Seitz symbolism, see Section 1.4.2.2 and Glazer et al. (2014).]
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 .
For magnetic group types , the magnetic group type symbol of the subgroup is given in the third column of the survey of magnetic groups, see e.g. Table 3.6.2.2. If is a group , then the subgroup is defined by the translational group of and the unprimed coset representatives of .
Example
Consider the threedimensional magnetic spacegroup type 16.3.101 . The representative group is defined by the translational subgroup denoted by the letter P generated by the translationsand the standard set of coset representativesThe subgroup of index 2 of the representative group is defined by the translational group denoted by the letter P and the cosets {1∣0} and {2_{001}∣0}, and is a group of type P2.
If is a group , then the subgroup is defined by the nonprimed translational group of and all the cosets of the standard set of coset representatives of the group .
Example
Consider the threedimensional magnetic spacegroup type 16.4.102 P_{2a}222. The representative group P_{2a}222 is defined by the translational group denoted by the symbol P_{2a} generated by the translationsand the standard set of coset representativesThe subgroup of index 2 of the representative group is defined by the translational subgroup denoted by the symbol P_{2a}, i.e. the translations generated byand the standard set of cosets of P_{2a}222. The group is a group of type P222.
While the group type symbol of is given, the coset representatives of the subgroup of derived from the standard set of coset representatives of may not be identical with the standard set of coset representatives of the representative group of type found in the survey of magnetic group types. Consequently, to show the relationship between this subgroup and the listed representative group of groups of type additional information is provided: a new coordinate system is defined in which the coset representatives of this subgroup are identical with the standard set of coset representatives listed for the representative group of groups of type : Let (O; a, b, c) be the coordinate system in which the group is defined. O is the origin of the coordinate system, and a, b and c are the basis vectors of the coordinate system. a, b and c represent a set of basis vectors of a primitive cell for primitive lattices and of a conventional cell for centred lattices. A second coordinate system, defined by (O + p; a′, b′, c′), is given in which the coset representatives of this subgroup are identical with the standard set of coset representatives listed for the representative group of groups of type . O + p is referred to as the location of the subgroup in the coordinate system of the group (Kopský, 2011). The origin is first translated from O to O + p. On translating the origin from O to O + p, a coset representative {R∣τ} becomes {R∣τ + Rp − p} (Litvin, 2005, 2008b; see also Section 1.5.2.3 ). This is followed by changing the basis vectors a, b and c to a′, b′ and c′, respectively. The basis vectors a′, b′, c′ define the conventional unit cell of the nonprimed subgroup of in the coordinate system (O; a, b, c) in which is defined. (O + p; a′, b′, c′) is given immediately following the group type symbol for the subgroup of . [In Litvin (2013), for typographical simplicity, the symbols `O +' are omitted.]
Example
For the threedimensional magnetic spacegroup type 10.4.52, , one finds in Litvin (2013)^{2}

The translational subgroup of the subgroup = P2 of is generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 1} and the coset representatives of this group are {1∣0} and {2_{010}∣0}, the unprimed coset representatives on the right. This subgroup is of type P2. In Litvin (2013), listed for the group type 3.1.8, P2, one finds the identical two coset representatives. Consequently, there is no change in the coordinate system, i.e. and a′ = a, b′ = b and c′ = c. In the coordinate system of the magnetic group , the coset representatives of its subgroup are identical with the standard set of coset representatives of the group type P2.
Example
For the threedimensional magnetic spacegroup type 16.7.105, one has

The translational subgroup of the subgroup of is generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 2}, and the coset representatives of this group are all those coset representatives on the right. This subgroup is of type P222_{1}. Listed for the group type 17.1.106 P222_{1}, one finds a different set of coset representatives:Consequently, to show the relationship between this subgroup of and the listed representative group of the group type P222_{1} we change the coordinate system in which is defined to (0, 0, 0; a, b, 2c). In this new coordinate system the coset representatives of the subgroup are identical with the coset representatives of the representative group of the group type P222_{1}.
Example
For the threedimensional magnetic spacegroup type 18.4.116, P2_{1}2_{1}′2′, one has

The translational subgroup of is generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 1} and the coset representatives of this group are {1∣0} and {2_{100}∣½, ½, 0}, the unprimed coset representatives on the right. The group is of type P2_{1}. For the magnetic group type 4.1.15 P2_{1} one finds a different set of coset representatives: {1∣0} and {2_{010}∣0, ½, 0}. Consequently, to show the relationship between the subgroup of and the listed representative group of the group type P2_{1}, we change the coordinate system in which the subgroup is defined to (0, ¼, 0; c, a, b). The origin is first translated from O to O + p, where p = (0, ¼, 0), and then a new set of basis vectors, a′ = c, b′ = a and c′ = b, is defined. In this new coordinate system the coset representatives of the subgroup are identical with the standard set of coset representatives of the representative group of the group type P2_{1}.
In this section we present a guide to the tables of properties of the two and threedimensional magnetic subperiodic groups and the one, two and threedimensional magnetic space groups given by Litvin (2013). The format and content of these magnetic group tables are similar to the format and content of the spacegroup tables in the present volume, the subperiodic group tables in IT E, and previous compilations of magnetic subperiodic groups (Litvin, 2005) and magnetic space groups (Litvin, 2008b). An example of one these tables is given in Fig. 3.6.3.1. The tables of properties of magnetic groups contain:
First page:
(1) Lattice diagram
(2) Headline
(3) Diagrams of symmetry elements and of the general positions
(4) Origin
(5) Asymmetric unit
(6) Symmetry operations
Subsequent pages:
(7) Abbreviated headline
(9) General and special positions with spins (magnetic moments)
(10) Symmetry of special projections
Tabulations of properties of threedimensional magnetic space groups can also be found in Koptsik (1968) (note that the generalposition diagrams are of `black and white' objects, not spins). Neutrondiffraction extinctions can be found in the work of Ozerov (1969a,b) and on the Bilbao Crystallographic Server, http://www.cryst.ehu.es (Aroyo et al., 2006). General positions and Wyckoff positions of the threedimensional magnetic space groups can also be found on the Bilbao Crystallographic Server.
For each threedimensional magnetic space group, a threedimensional lattice diagram is given in the upper lefthand corner of the first page of the tables of properties of that group. (For all other magnetic groups, the corresponding lattice diagram is given within the symmetry diagram, see Section 3.6.3.3 below.) This lattice diagram depicts the coordinate system used, the conventional unit cell of the space group , the magnetic space group's magnetic superfamily type and the generators of the translational subgroup of the magnetic space group. In Fig. 3.6.3.2 we show lattice diagrams for two orthorhombic magnetic space groups: (a) Pmc2_{1} and (b) P_{2b}m′c′2_{1}. The generating lattice vectors are colour coded. Those coloured black are not coupled with time inversion, while those coloured red are coupled with time inversion. In the group Pmc2_{1}, a magnetic group of the type , the lattice is an orthorhombic P lattice, see Fig. 3.6.3.2(a), and no generating translation is coupled with time inversion. In the second group, P_{2b}m′c′2_{1}, a magnetic group of type , the lattice is an orthorhombic P_{2b} lattice, see Fig. 3.6.3.2(b), and the generating lattice vector in the y direction is coupled with time inversion.
Each table begins with a headline consisting of two lines with five entries, for exampleFor threedimensional magnetic space groups, this headline is to the right of the lattice diagram. On the upper line, starting on the left, are three entries:
The second line has two additional entries:
There are two types of diagrams: symmetry diagrams and generalposition diagrams. The symmetry diagrams show (1) the relative locations and orientations of the symmetry elements and (2) the absolute locations and orientations of these symmetry elements in a given coordinate system. The generalposition diagrams show, in that coordinate system, the arrangement of a set of symmetryequivalent general points and the relative orientations of magnetic moments on this set of points. Figs. 3.6.3.3 and 3.6.3.4 show the symmetry diagram and generalposition diagram, respectively, of the threedimensional magnetic space group P4_{1}2′2′.
All diagrams of threedimensional magnetic space groups and threedimensional subperiodic groups are orthogonal projections. The projection direction is along a basis vector of the conventional crystallographic coordinate system, see Table 1.1 of Litvin (2013). If the other two basis vectors are not parallel to the plane of the diagram, they are indicated by a subscript p, e.g. a_{p}, b_{p} and c_{p}. Schematic representations of the diagrams, showing their conventional coordinate systems, i.e. the origin O and basis vectors, are given in Table 2.1 of Litvin (2013). For twodimensional magnetic space groups and magnetic frieze groups, the diagrams are in the plane defined by the group's conventional coordinate system.
The graphical symbols used in the symmetry diagrams are listed in Table 2.2 of Litvin (2013) and are an extension of those used in the present volume, IT E and Litvin (2008b). For symmetry planes and symmetry axes parallel to the plane of diagram, for rotationinversions and for centres of symmetry, the `heights' h along the projection direction above the plane of the diagram are given. The heights are given as fractions of the shortest translation along the projection direction and, if different from zero, are printed next to the graphical symbol, see Fig. 3.6.3.3.
In the generalposition diagrams, the general positions and corresponding magnetic moments are colour coded. Positions with a z component of +z are shown as red circles and those with a z component of −z are shown as blue circles. If the z component is either h + z or h − z with h ≠ 0, then the height h is printed next to the general position, see Fig. 3.6.3.4. If two general positions have the same x component and y component, but one has a z component +z and the other −z, the positions are shown as a circle with one half coloured red, the other half blue. The magnetic moments are colour coded to the general position to which they are associated, their direction in the plane of projection is given by an arrow in the direction of the projected magnetic moment. A + or − sign near the tip of the arrow indicates that the magnetic moment is inclined, respectively, above or below the plane of projection.
For magnetic space groups of the type , the symmetry diagram is that of the group . That each symmetry element also appears coupled with time inversion is represented by a red printed between and above the generalposition and symmetry diagrams. Because groups of this type contain the timeinversion symmetry, the magnetic moments are all identically zero, and no arrows appear in the generalposition diagram. An example, the diagrams of the magnetic space group P4_{1}221′ are shown in Fig. 3.6.3.5. For triclinic, monoclinic/oblique, monoclinic/rectangular and orthorhombic rod groups, the colour coding of the general positions is extended according to the positive or negative values of the x and z components of the coordinates of the general position, see Fig. 3.6.3.6.

Generalposition diagram of rod group 2.3.29 P2/c′11. The positional colour coding is red for x > 0 and z > 0; blue for x > 0 and z < 0; green for x < 0 and z > 0; and brown for x < 0 and z < 0. 
VRML (Virtual Reality Modeling Language) generalposition diagrams are available for the two and threedimensional magnetic subperiodic groups (Cordisco & Litvin, 2004), and for the one, two and noncubic threedimensional magnetic space groups (Burke et al., 2006). These diagrams can be rotated and zoomed in on to aid in the visualization of the generalposition diagrams, and include both the general positions of the atoms and the general orientations of the associated magnetic moments.
If the magnetic space group is centrosymmetric, then the inversion centre or a position of high site symmetry, as on the fourfold axis of tetragonal groups, is chosen as the origin. For noncentrosymmetric groups, the origin is at a point of highest site symmetry. If no site symmetry is higher than 1, the origin is placed on a screw axis, a glide plane or at the intersection of several such symmetries.
In the Origin line below the diagrams, the site symmetry of the origin is given. An additional symbol indicates all symmetry elements that pass through the origin. For example, for the magnetic space group I4/mcm, one finds `Origin at center (4/m) at 4/mc2_{1}/c'. The site symmetry is 4/m and, in addition, two glide planes perpendicular to the y and z axes, and a screw axis parallel to the z axis, pass through the origin.
An asymmetric unit is a simply connected smallest part of space which, by application of all symmetry operations of the magnetic group, exactly fills the whole space. For subperiodic groups, because the translational symmetry is of a lower dimension than that of the space, the asymmetric unit is infinite in size. The asymmetric unit for subperiodic groups is defined by setting the limits on the coordinates of points contained in the asymmetric unit. For example, the asymmetric unit for the magnetic layer group 32.3.199 pm′2_{1}n′ isSince the translational symmetry of a magnetic space group is of the same dimension as that of the space, the asymmetric unit is a finite part of space. The asymmetric unit is defined, as above, by setting the limits on the coordinates of points contained in the asymmetric unit. For example, for the magnetic space group 140.3.1198 I4/m′cm one hasDrawings showing the boundary planes occurring in the tetragonal, trigonal and hexagonal systems, together with their algebraic equations, are given in Fig. 2.1.3.11 . Drawings of asymmetric units for cubic groups have been published by Koch & Fischer (1974). The asymmetric units have complicated shapes in the trigonal, hexagonal and cubic crystal systems, and consequently are also specified by giving the vertices of the asymmetric unit. For example, for the magnetic space group 176.1.1374 P6_{3}/m one finds

Because the asymmetric unit is invariant under time inversion, all magnetic space groups , and of the magnetic superfamily of type have identical asymmetric units, the asymmetric unit of the group (as in the present volume).
Listed under the heading of Symmetry operations is the geometric description of the symmetry operations of the magnetic group. A symbol denoting the geometric description of each symmetry operation is given. Details of this symbolism, except for the use of prime to denote time inversion, are given in Sections 1.4.2 and 2.1.3.9 . For glide planes and screw axes, the glide and screw part are always explicitly given in parentheses by fractional coordinates, i.e. by fractions of the basis vectors of the coordinate system of of the superfamily of the magnetic group. A coordinate triplet indicating the location and orientation of the symmetry element is given, and for rotationinversions the location of the inversion point is also given. These symbols, with the addition of a prime to denote time inversion, follow those used in the present volume, IT E and Litvin (2005, 2008b). In addition, each symmetry operation is also given in Seitz (1934, 1935a,b, 1936) notation (see Section 3.6.2.2.3), e.g. see Table 3.6.3.1 for the symmetry operations of the magnetic space group 51.14.400 P_{2b}mma′.

The corresponding coordinate triplets of the General positions, see Section 3.6.3.9, may be interpreted as a second description of the symmetry operations, a description in matrix form. The numbering (1), (2), …, (p), … of the entries in the blocks Symmetry operations is the same as the numbering of the corresponding coordinate triplets of the General position, the first block below Positions. For all magnetic groups with primitive lattices, the two lists, Symmetry operations and General position, have the same number of entries.
For magnetic groups with centred cells, only one block of several (two, three or four) blocks of the general positions is explicitly given, see Table 3.6.3.2. A set of two, three or four centring translations is given below the subheading Coordinates. Each of these translations is added to the given block of general positions to obtain the complete set of blocks of general positions. While one of the several blocks of general positions is explicitly given, the corresponding symmetry operations are all explicitly given. Each corresponding block of symmetry operations is listed under a subheading of `centring translation + set' for each centring translation listed below the subheading Coordinates.

On the second and subsequent pages of the tables for a specific magnetic group there is an abbreviated headline. This abbreviated headline contains three items: (1) the word `Continued', (2) the threepart number of the magnetic group type, and (3) the short international (Hermann–Mauguin) symbol for the magnetic group type.
The line Generators selected lists the symmetry operations selected to generate the symmetryequivalent points of the General position from a point with coordinates x, y, z. The first generator is always the identity operation given by (1) followed by generating translations. Additional generators are given as numbers (p), which refer to the coordinate triplets of the General position and to corresponding symmetry operations in the first block, if more than one, of the Symmetry operations.
The entries under Positions, referred to as Wyckoff positions, consist of the General position, the upper block, followed by blocks of Special positions. The upper block of positions, the general position, is a set of symmetryequivalent points where each point is left invariant only by the identity operation or, for magnetic groups , by the identity operation and time inversion, but by no other symmetry operations of the magnetic group. The lower blocks, the special positions, are sets of symmetryequivalent points where each point is left invariant by at least one additional operation in addition to the identity operation, or, for magnetic space groups , in addition to the identity operation and time inversion.
For each block of positions the following information is provided:
Multiplicity: The multiplicity is the number of equivalent positions in the conventional unit cell of the nonprimed group associated with the magnetic group.
Wyckoff letter: This letter is a coding scheme for the blocks of positions, starting with `a' at the bottom block and continuing upwards in alphabetical order.
Site symmetry: The sitesymmetry group is the largest subgroup of the magnetic space group that leaves invariant the first position in each block of positions. This group is isomorphic to a subgroup of the point group of the magnetic group. An `oriented' symbol is used to show how the symmetry elements at a site are related to the conventional crystallographic basis, and the sequence of characters in the symbol correspond to the sequence of symmetry directions in the magnetic group symbol. Sets of equivalent symmetry directions that do not contribute any element to the site symmetry are represented by dots. Sets of symmetry directions having more than one equivalent direction may require more than one character if the sitesymmetry group belongs to a lower crystal system. For example, for the 2c position of the magnetic space group P4′m′m (99.3.825) the sitesymmetry group is `2m′m′.'. The two characters m′m′ represent the secondary set of tetragonal symmetry directions, whereas the dot represents the tertiary tetragonal symmetry directions.
Coordinates of positions and components of magnetic moments: In each block of positions, the coordinates of each position are given. Immediately following each set of position coordinates are the components of the symmetryallowed magnetic moment at that position. The components of the magnetic moment of the first position are determined from the given sitesymmetry group. The components of the magnetic moments at the remaining positions are determined by applying the symmetry operations to the components of that magnetic moment at the first position.
The symmetry of special projections is given for the two and threedimensional magnetic groups. For each threedimensional magnetic group, the symmetry is given for three projections, projections onto planes normal to the projection directions. If there are three symmetry directions, the three projection directions correspond to primary, secondary and tertiary symmetry directions. If there are fewer than three symmetry directions, the additional projection direction or directions are taken along coordinate axes. For twodimensional magnetic groups, there are two orthogonal projections. The projections are onto lines normal to the projection directions.
For the threedimensional magnetic space groups, each projection gives rise to a twodimensional magnetic space group. For twodimensional magnetic space groups, each projection gives rise to a onedimensional magnetic space group. For magnetic rod groups and magnetic layer groups, a projection along the [001] direction gives rise, respectively, to a twodimensional magnetic point group and a twodimensional magnetic space group. All other projections give rise to magnetic frieze groups. For magnetic frieze groups, projections give rise to either a onedimensional magnetic space group or a onedimensional magnetic point group. The international (Hermann–Mauguin) symbol of the symmetry group of each projection is given. Below this symbol, the basis vector(s) of the projected symmetry group and the origin of the projected symmetry group are given in terms of the basis vector(s) of the projected magnetic group. The location of the origin of the symmetry group of the projection is given with respect to the unit cell of the magnetic group from which it has been projected.
There are other notations for magnetic group type symbols than the notations of Opechowski & Guccione (1965) and Belov, Neronova & Smirnova (1957): for example for the threedimensional magnetic group 55.10.450 P_{2c}b′a′m the Shubnikov notation is (Koptsik, 1966; Shubnikov & Koptsik, 1974) or (Ozerov, 1969a,b) (see also Zamorzaev, 1976). There are also the variations of the Opechowski & Guccione notation put forward by Grimmer (2009, 2010). We shall limit ourselves here to a detailed comparison of the Opechowski & Guccione and Belov, Neronova & Smirnova notations.
For all group types in the reduced magnetic superfamily of , the Opechowski & Guccione (1965) magnetic group type symbols (OG symbols) are based on the symbol of the group . Belov, Neronova & Smirnova (1957) also base their symbols (BNS symbols) on the symbol of the group , but only for magnetic groups of the type , and . For magnetic groups , where is an equiclass subgroup of , the BNS symbol is based on the symbol of the group , the nonprimed subgroup of index 2. A magnetic group can be written as , where is a translation of not in . The BNS symbol for a magnetic group of the type is the symbol for the group type with a subindex inserted on the symbol for the translational subgroup of to denote the translation .
Example
The representative threedimensional space group has a translational subgroup generated by the three translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 1} and the standard set of coset representativesThe threedimensional magnetic space group 25.10.165 = has a subgroup with a translational subgroup generated by the three translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 2}, = {1∣0, 0, 1}′, and a set of coset representativesThe OG magnetic group type symbol is, see Section 3.6.2.2.4, P_{2c}m′m′2, i.e. based on the symbol for the group type = . The BNS symbol is P_{c}cc2, i.e. based on the symbol for the subgroup of , with a subindex `c' attached to `P' to denote the translation = {1∣0, 0, 1}′ in = .
A sidebyside comparison of OG magnetic group type symbols and BNS symbols is given in Litvin (2013). As the OG and BNS symbols are the same for magnetic groups , and , BNS symbols are explicitly listed only for groups of type . Examples of this comparison are given in Table 3.6.4.1.

We consider the maximal subgroups of index ≤ 4 of the one, two and threedimensional magnetic space groups and the two and threedimensional magnetic subperiodic groups. A complete listing of the maximal subgroups of the two and threedimensional nonprimed space groups can be found in International Tables for Crystallography, Volume A1, Symmetry Relations Between Space Groups (2010; IT A1). The maximal subgroups of index ≤ 4 of the threedimensional nonprimed space groups and nonprimed layer and rod groups can also be found on the Bilbao Crystallographic Server (http://www.cryst.ehu.es ; Aroyo et al., 2006).
For magnetic groups, an abstract of a method for determining the maximal subgroups of magnetic groups was published by Sayari & Billiet (1977). The maximal subgroups of magnetic groups found in Litvin (2013) were derived from Litvin (2008a) using a method given by Litvin (1996).
Each maximal subgroup table is headed by the magnetic group type whose maximal subgroup types are to be listed.
Examples
For the threedimensional magnetic space group type , one finds (Litvin, 2013), in bold blue type, information which defines the representative group of this type in a coordinate system O; a, b, c:The first column gives the global serial number of the group, followed in the second column by its magnetic group type symbol. In the third column, the symbol (O; a′, b′, c′) gives the origin O and basis vectors a′, b′, c′ of the conventional unit cell of the nonprimed subgroup of the representative group of the type Pb′a′m. These basis vectors a′, b′, c′ imply both the magnetic and nonprimed translational subgroups of the representative group. In this case, the P translational subgroup of the representative group is nonprimed and generated by the basis vectors a, b, c. The standard set of coset representatives of this representative group is given on the right.
For the threedimensional magnetic spacegroup type P_{2b}ma2 one finds (Litvin, 2013):Note here that a′ = a, b′ = 2b, c′ = c, being the basis vectors of the conventional unit cell of the nonprimed subgroup of the representative group of P_{2b}ma2, implies that the translational subgroup of the representative group is P_{2b}, i.e. generated by the nonprimed translations {1∣1, 0, 0}, {1∣0, 2, 0}, {1∣0, 0, 1} and the magnetic translation {1∣0, 1, 0}′.
Following the subtable heading of each magnetic spacegroup type is a listing of the maximal subgroups of index ≤ 4 of the representative magnetic space group of this type.
Examples
From the list of maximal subgroups of the representative magnetic group of the type Pb′a′m is the subgroup listed asThe first column gives the magnetic group type symbol of the subgroup. The second column gives the subgroup index of this subgroup as a subgroup of the representative group of type Pb′a′m. In the third column, the symbol (O + p; a′, b′, c′) gives the origin `O + p' and basis vectors a′, b′, c′ of the conventional unit cell of the nonprimed subgroup, where p, a translation of the origin of the coordinate system of the representative group Pb′a′m, and the conventional unit cell are such that the coset representatives listed are transformed into the standard cosets of the representative group of the subgroup type Pb′a′2. In this case, since the listed coset representatives are the standard cosets of the representative group of type Pb′a′2, no translation of origin is required, and consequently O + p = 0, 0, 0. The conventional unit cell of the nonprimed subgroup of the subgroup Pb′a′2 is the same as that of representative group of the type Pb′a′2 and consequently one finds a′ = a, b′ = b, c′ = c.
That the listed coset representatives of the subgroup are the same as those of the representative group of that subgroup type is not always the case:
Example
A subgroup of type P2/m of the representative group Pb′a′m is listed asThe standard set of coset representatives of the representative group P2/m areA change in setting to have the coset representatives of the subgroup be identical with the coset representatives of the representative group P2/m is represented in the symbol (0, 0, 0; b, c, a), i.e. changing the setting from a, b, c to b, c, a.
Other cases may require a simultaneous change to both the origin and the setting of the conventional unit cell of the nonprimed subgroup:
Example
A third subgroup of the representative group Pb′a′m is the equiclass subgroup of the same type Pb′a′m:The listed coset representatives of this subgroup are not the same as the coset representatives of the representative group Pb′a′m, i.e. where the z component of the nonprimitive translation associated with all coset representatives is zero. To have these listed coset representatives become identical with the coset representatives of the standard representative group of Pb′a′m, one must change the origin of the coordinate system. This information is provided in the symbol (0, 0, ½; a, b, 2c) where O + p = 0, 0, ½ denotes the translation under which all the nonzero z components of the coset representatives are transformed to zero. Note also that the P in the subgroup symbol denotes a nonprimed translational subgroup which is determined by a′ = a, b′ = b, c′ = 2c, i.e. P denotes the translational group generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 2}.
In the tabulations of the maximal subgroups of groups of the type not all maximal subgroups are explicitly listed. The maximal subgroup of is not listed. If is a maximal subgroup of , then is a maximal subgroup of and is also not explicitly listed. All maximal subgroups of are listed under , and consequently, all maximal subgroups of are then found from the list of all maximal subgroups of by multiplying each by . For each listed maximal subgroup, its nonprimed subgroup type is explicitly given. For example, a listed subgroup of Pma21′ iswhere the nonprimed subgroup type Pm of Pma′2′ is given in the second column.
References
Aroyo, M. I., PerezMato, J. M., Capillas, C., Kroumova, E., Ivantchev, S., Madariaga, G., Kirov, A. & Wondratschek, H. (2006). Bilbao Crystallographic Server I: Databases and crystallographic computing programs. Z. Kristallogr. 221, 15–27.Belov, N. V. (1956a). The onedimensional infinite crystallographic groups. Kristallografiya, 1, 474–476.
Belov, N. V. (1956b). The onedimensional infinite crystallographic groups. Sov. Phys. Crystallogr. 1, 372–374.
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.
Bradley, C. J. & Cracknell, A. P. (1972). The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
Burke, J. S., Cordisco, N. R. & Litvin, D. B. (2006). VRML general position diagrams of magnetic space groups. J. Appl. Cryst. 39, 620.
Cochran, W. (1952). The symmetry of real periodic twodimensional functions. Acta Cryst. 5, 630–633.
Cordisco, N. R. & Litvin, D. B. (2004). VRML general position diagrams of the magnetic subperiodic groups. J. Appl. Cryst. 37, 346.
Galyarskii, E. I. & Zamorzaev, A. M. (1965a). A complete description of crystallographic stem groups of symmetry and different types of antisymmetry. Kristallografiya, 10, 147–154.
Galyarskii, E. I. & Zamorzaev, A. M. (1965b). A complete description of crystallographic stem groups of symmetry and different types of antisymmetry. Sov. Phys. Crystallogr. 10, 109–115.
Glazer, A. M., Aroyo, M. I. & Authier, A. (2014). Acta Cryst. A70, 300–302.
Grimmer, H. (2009). Comments on tables of magnetic space groups. Acta Cryst. A65, 145–155.
Grimmer, H. (2010). Opechowski–Guccionelike symbols labelling magnetic space groups independent of tabulated (0, 0, 0)+ sets. Acta Cryst. A66, 284–291.
Guccione, R. (1963a). Magnetic Space Groups. PhD Thesis, University of British Columbia, Canada.
Guccione, R. (1963b). On the construction of the magnetic space groups. Phys. Lett. 5, 105–107.
Heesch, H. (1929). Zur strukturtheorie der ebenen symmetriegruppen. Z. Kristallogr. 71, 95–102.
Heesch, H. (1930). Uber die vierdimensionalen gruppen des dreidimensionalen raumes. Z. Kristallogr. 73, 325–345.
International Tables for Crystallography (2010). Vol. A1, Symmetry Relations Between Space Groups, 2nd ed., edited by H. Wondratschek & U. Müller. Chichester: Wiley. [First edition 2004.]
International Tables for Crystallography (2010). Vol. E, Subperiodic Groups, 2nd ed., edited by V. Kopský & D. B. Litvin. Chichester: Wiley. [First edition 2002.]
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.]
Koch, E. & Fischer, W. (1974). Zur Bestimmung asymmetrischer Einheiten kubischer Raumgruppen mit Hilfe von Wirkungsbereichen. Acta Cryst. A30, 490–496.
Kopský, V. (2011). Private communication.
Koptsik, V. A. (1966). Shubnikov Groups. Handbook on the Symmetry and Physical Properties of Crystal Structures. Izd. MGU (in Russian). English translation of text: Kopecky, J. & Loopstra, B. O. (1971). Fysica Memo 175. Stichting, Reactor Centrum Nederland.
Koptsik, V. A. (1967). A general sketch of the development of the theory of symmetry and its applications in physical crystallography over the past 50 years. Kristallografiya, 12, 755–774.
Koptsik, V. A. (1968). A general sketch of the development of the theory of symmetry and its applications in physical crystallography over the past 50 years. Sov. Phys. Crystallogr. 12, 667–683.
Landau, L. I. & Lifschitz, E. M. (1951). Statistical Physics. Moscow: Gostekhizdat (in Russian); English translation (1958). Oxford: Pergamon Press.
Landau, L. I. & Lifschitz, E. M. (1957). Electrodynamics of Continuous Media. Moscow: Gostekhizdat (in Russian); English translation (1960). Reading, MA: AddisonWesley.
Litvin, D. B. (1996). Maximal subgroups of magnetic space groups and subperiodic groups. Acta Cryst. A52, 681–685.
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. (1999). Magnetic subperiodic groups. Acta Cryst. A55, 963–964.
Litvin, D. B. (2001). Magnetic spacegroup types. Acta Cryst. A57, 729–730.
Litvin, D. B. (2005). Tables of properties of magnetic subperiodic groups. Acta Cryst. A61, 382–385.
Litvin, D. B. (2008a). Magnetic group maximal subgroups of index 4. Acta Cryst. A64, 345–347.
Litvin, D. B. (2008b). Tables of crystallographic properties of magnetic space groups. Acta Cryst. A64, 419–424.
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 .
Neronova, N. N. & Belov, N. V. (1961a). A single scheme for the classical and blackandwhite crystallographic symmetry groups. Kristallografiya, 6, 3–12.
Neronova, N. N. & Belov, N. V. (1961b). A single scheme for the classical and blackandwhite crystallographic symmetry groups. Sov. Phys. Crystallogr. 6, 1–9.
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.
Ozerov, R. P. (1969a). The design of tables of Shubnikov space groups of dichroic symmetry. Kristallografiya, 14, 393–403.
Ozerov, R. P. (1969b). The design of tables of Shubnikov space groups of dichroic symmetry. Sov. Phys. Crystallogr. 14, 323–332.
Palistrant, A. F. & Zamorzaev, A. M. (1964a). Groups of symmetry and different types of antisymmetry of borders and ribbons. Kristallografiya, 9, 155–161.
Palistrant, A. F. & Zamorzaev, A. M. (1964b). Groups of symmetry and different types of antisymmetry of borders and ribbons. Sov. Phys. Crystallogr. 9, 123–128.
Sayari, A. & Billiet, Y. (1977). Subgroups and changes of standard setting of triclinic and monoclinic space groups. Acta Cryst. A33, 985–986.
Seitz, F. Z. (1934). A matrixalgebraic development of the crystallographic groups. Z. Kristallogr. 88, 433–459.
Seitz, F. Z. (1935a). A matrixalgebraic development of the crystallographic groups. Z. Kristallogr. 90, 289–313.
Seitz, F. Z. (1935b). A matrixalgebraic development of the crystallographic groups. Z. Kristallogr. 91, 336–366.
Seitz, F. Z. (1936). A matrixalgebraic development of the crystallographic groups. Z. Kristallogr. 94, 100–130.
Shubnikov, A. V. (1945). New ideas in the theory of symmetry and its applications. In Report of the General Assembly of the Academy of Sciences of the USSR, 14–17 October 1944. Izd. Acad. Nauk SSSR, 212–227.
Shubnikov, A. V. (1959a). Symmetry and antisymmetry of rods and semicontinua with principal axis of infinite order and finite transfers along it. Kristallografiya, 4, 279–285.
Shubnikov, A. V. (1959b). Symmetry and antisymmetry of rods and semicontinua with principal axis of infinite order and finite transfers along it. Sov. Phys. Crystallogr. 4, 262–266.
Shubnikov, A. V., Belov, N. V. & others (1964). Colored Symmetry, London: Pergamon Press.
Shubnikov, A. V. & Koptsik, V. A. (1974). Symmetry in Science and Art. New York: Plenum Press.
Stokes, H. T. & Campbell, B. J. (2009). Table of magnetic space groups, http://stokes.byu.edu/magneticspacegroupshelp.html.
Zamorzaev, A. M. (1953). Dissertation, Leningrad State University, Russia. (In Russian.)
Zamorzaev, A. M. (1957a). Generalizations of Fedorov groups. Kristallografiya, 2, 15–20.
Zamorzaev, A. M. (1957b). Generalizations of Fedorov groups. Sov. Phys. Crystallogr. 2, 10–15.
Zamorzaev, A. M. (1976). The Theory of Simple and Multiple Antisymmetry. Kishinev: Shtiintsa. (In Russian.)
Zamorzaev, A. M. & Palistrant, A. P. (1980). Antisymmetry, its generalizations and geometrical applications. Z. Kristallogr. 151, 231–248.