A More Rational Order for the 230 Space Groups

This project began in 2023 with two simple questions:

Why is space group C222₁ (#20) before space group C222 (#21) in Vol. A of International Tables for Crystallography (2016)?

Why are the Pn_m, n = 4 and 6 space groups ordered differently in that volume? (i.e., P4₁, P4₂, P4₃ vs. P6₁, P6₅, P6₂, P6₄, P6₃)

The resulting paper has now appeared in the Teaching Section of the Journal of Applied Crystallography (Aroyo & Brock, 2026). That paper outlines the history of the ordering of the space groups (with hyperlinks for some of the original books), lists the very great number of inconsistencies in the way the groups are currently ordered, and proposes an alternative order.

That longish paper is accompanied by an even longer Supplementary Commentary providing more complete explanations of some points, along with many things we found interesting but not central. These extra items include Sohncke’s drawing (Sohncke, 1879) of his space group #13, which is not a group (Fig. 1), and a comparison of the orderings of Schoenflies (1891) and Fedorov (1891).

The short answer to the questions above is that the order within the geometric crystal classes (commonly referred to as the point-group classes) is that of Schoenflies (1891). Perhaps he was satisfied with the very major accomplishment of having found, by himself and then later in collaboration with Fedorov, the complete set of crystallographic space groups, and then wanted to move on to other projects rather than developing rules for ordering the groups. There are fewer inconsistencies in Fedorov’s (1891) ordering, but it was not adopted, possibly because his notation is difficult and possibly because German was more widely understood than Russian. In 1919 Paul Niggli proposed a more logical order (Niggli, 1919), but noted that it was too late for a change. We agree, partly because any revised order would change the number of space group #14 (P2₁/c, etc.), which is deeply embedded in many crystallographers’ minds. Space-group numbers are, however, just attributes, and there could be two sets in the electronic versions of Vols. A (2016) and A1 (2011) of the series International Tables for Crystallography.

The order of the space groups depends on the order of the crystal systems, the order of the geometric crystal classes, and the rules (or lack thereof) for determining the order within those classes. Committees of the IUCr have argued for decades about the identities of the seven (or six) crystal systems and their order; the choices made seem final. The order of the geometric crystal classes (Fig. 2) has not changed since 1952 (International Tables for Crystallography, 1952) and has not been questioned recently. The problem then is with the ordering of the groups within the geometric crystal classes. It seems strange that there are so many inconsistencies in a tabulation published by a scientific community that cares so much about order.

The key rules for sorting within a geometric crystal class follow; for details see Aroyo & Brock, 2026.

1. Groups are sorted first by arithmetic crystal class, which is the point group followed by the centering type in the order P > C > A > F > I and P > R, where > means before.
   While this order is generally followed in Vol. A, there are exceptions (Fig. 3), and the 432P, 432F, and 432I groups are interspersed. This rule necessarily changes the number of space group #14.

2. Within each arithmetic crystal class the symmorphic group is first. [A symmorphic group has no screw axis or glide plane in its full Hermann-Mauguin (hereafter, HM) symbol.]
   This rule reverses the order of C222₁ and C222.

3. The remaining groups in the arithmetic crystal class are sorted by the extent to which they differ from the symmorphic group, i.e., by the number of screw axes and glide planes that appear in the full HM symbol. Following Niggli’s (1919) lead, all groups having a screw axis in the full HM symbol follow all groups having none.

4. For sets of groups that differ only by the value m of a screw axis n_m, members of a pair of enantiomorphic groups are adjacent. Groups that have rotation axes coincident with the screw axes precede groups that do not.
   The orders of the Pn_m groups are then P4₂, P4₁, P4₃ for n = 4 and P6₃, P6₂, P6₄, P6₁, P6₅ for n = 6.

5. Groups having glide planes in the full HM symbol are sorted according to the existing priority order e > a > b > c > n.

The choices of axes for the space groups were established in 1935 (Internationale Tabellen zur Bestimmung von Kristallstrukturen, 1935). Those choices have not changed since except for the addition of the c-unique setting of the monoclinic groups. The logic behind the choices of the standard settings of the orthorhombic groups is unclear. The group Pna2₁, for example, could have been chosen as any one of six possibilities (Pbn2₁, Pn2₁a, P2₁cn, etc.; see Table 1.5.4.4 of the 2016 edition of Vol. A). Choosing a different setting as standard for a group would alter the proposed space-group ordering and might have some advantages, but trying to change the standard setting of any space group now seems unwise if not foolhardy.

An advantage of the proposed ordering is that it makes the relationships of the space groups more obvious. Students may wonder how Schoenflies, Fedorov, and Sohncke developed their lists of space groups. The answer is that they started with low-symmetry groups and then worked out where to add a symmetry element so that the addition of the corresponding symmetry operation(s) gives a new mathematical group. Schoenflies outlined the steps in his 1891 book. The proposed new sequence of space groups makes that process more obvious by putting the sets of space-group generators in a more rational order. The generators are a set of symmetry operations that can be combined to generate all the other symmetry operations of the space group.

We hope that this proposal will stimulate discussion and will be useful in teaching, and that the alternative ordering can be added as an option to Vols. A and A1.

References

Aroyo, M. I. & Brock, C. P. (2026). J. Appl. Cryst. 59, 932-942. https://doi.org/10.1107/S1600576726002360.

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). I. Band edited by C. Hermann. Berlin: Borntraeger.

International Tables for X-ray Crystallography (1952). Vol. I, Symmetry groups, 1st ed. edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.

International Tables for Crystallography (2016). Vol. A, Space-group symmetry 6th ed., edited by M. I. Aroyo. Chichester: Wiley. (and 2nd online edition, https://doi.org/10.1107/97809553602060000114).

International Tables for Crystallography (2011). Volume A1, Symmetry relations between space groups 2nd online ed., edited by H. Wondratschek & U. Müller. https://doi.org/10.1107/97809553602060000110

Niggli, P. (1919). Geometrische Kristallographie des Diskontinuums. Leipzig: Borntraeger. https://books.google.com/books?id=7w8mAQAAIAAJ

Schoenflies, A. (1891). Krystallsysteme und Krystallstruktur. Leipzig: Teubner. https://books.google.com/books?id=EjIjcgAACAAJ

Sohncke, L. (1879). Entwicklung einer Theorie der Kristallstruktur. Leipzig: Teubner. https://books.google.com/books?id=MOXZMgEACAAJ

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