Spacegroup Data Explorer

This website explores the crystallographic space-group and point-group datasets generated for httk. The pages use the JSON-LD datasets symmetry_basics.json.gz and transformations_hm_entry.json.gz, generated for this site. The copied transformations_std.json.gz provides the corresponding IT-standard data for download.

Deep linking spacegroup data

Individual pages can be linked directly by several stable, site-relative URL schemes (prepend the site base URL):

• /hall/<hall entry>/ — the canonical page for a specific Hall setting, e.g. /hall/p_1/ or /hall/-c_2a_2ac/. The Hall entry is the Hall symbol written in lower case with spaces replaced by underscores.
• /itn/<IT number>/ — the standard (ITA reference) setting of a space-group number, e.g. /itn/68/, /itn/227/.
• /nc/<n:c>/ — a specific conventional setting identified by its International Tables setting code n:c, e.g. /nc/68:1/, /nc/68:1ba-c/, /nc/146:H/.
• /hm/<HM entry>/ — a specific conventional setting identified by its HM entry, written with spaces removed and "/" replaced by ":", e.g. /hm/Ccca:1/, /hm/C2:m11/, /hm/P-1/.

The /itn/, /nc/ and /hm/ links redirect to the corresponding /hall/ page. Because the three equivalent #68 setting pairs share a single page, both conventional labels of a pair (both n:c codes and both HM entries) link to that same page.

Things good to know about space groups

• There are 230 distinct three-dimensional space-group types. They are commonly identified by the sequence number in which they appear in the International Tables for Crystallography (IT), 1−230.
• Different axis choices, origin choices, and cell choices of the space-group types are called "settings". There are many different representations identifying the space groups and their settings. The most well-known are:
  • Hermann–Mauguin (HM) symbols of which there are several variants (see below).
  • Hall symbols encode generators together with lattice and origin conventions, fixing a space group in a particular coordinate system. Different valid Hall symbols can generate the same group in the same setting.
• Specifically, a set of 530 "conventional spacegroup settings" are present in the IT tables. However, of these actually there are 527 distinct Hall entries (distinct operation sets in the recorded coordinate systems). Three pairs of conventional settings, all in spacegroup #68, are equivalent (and, hence, share the same Hall symbol):
  • C_c_c_a:1 (#68:1) and C_c_c_b:1 (#68:1ba-c) -- Hall symbol: C 2 2 -1ac
  • A_b_a_a:1 (#68:1cab) and A_c_a_a:1 (#68:1-cba) -- Hall symbol: A 2 2 -1ab
  • B_b_c_b:1 (#68:1bca) and B_b_a_b:1 (#68:1a-cb) -- Hall symbol: B 2 2 -1ab
  The httk symmetry-basics tables therefore contain 527 entries, one per distinct Hall entry. This does not count all possible settings, of which there are infinitely many. However, in some contexts the distinction between these equivalent settings matters, e.g., when discussing the mapping of Wyckoff positions in a transformation from one setting to another; in those cases we have 530 items usually keyed on the HM entry (see below).
• Variants of the HM symbols
  • Short: The compact notation used in IT Volume A, suppressing redundant symmetry directions. This explorer gives both the setting-specific and IT-standard forms.
  • Full: The expanded notation used in IT Volume A, retaining direction-specific symmetry information omitted from the short form.
  • Extended: The notation in IT Volume A that supplies additional symmetry-element information, sometimes over multiple aligned lines; it is less commonly used than short or full symbols.
  • Entry:The list of all 530 conventional settings in IT table vol B, A1.4.2.7 includes a column titled "HM entry" which is the same as the full HM symbol, but where the older glide-plane letters are used, rather than the newer "e" notation (see below), and a disambiguation suffix is added if needed to distinguish between the 530 conventional space group settings.
  • Universal: The well-known Computational Crystallography Toolbox (cctbx) defines a "universal" variant with a (somewhat different) disambiguating suffix.
• The IT HM symbols were changed in the Fourth Edition of the IT tables (IT 1995) to replace the glide plane symbol: "For the five space groups Aem2 (39), Aea2 (41), Cmce (64), Cmme (67) and Ccce (68), the ‘new’ space-group symbols, containing the symbol ‘e’ for the ‘double’ glide plane, are given for all settings. These symbols were first introduced in the Fourth Edition of this volume (IT 1995); cf. Foreword to the Fourth Edition. For further explanations, see Section 1.3.2, Note (x) and the space-group diagrams". These older symbols are included as aliases in these tables.
• Hall-symbol spellings also vary between the original Hall tables and IT Volume B (2001). In particular, IT Volume B drops the original lattice symbol "S" and replaces some symbols with equivalent alternatives; see the SgInfo notes on Hall symbols. This historical "S" should not be confused with the setting-independent side-centred Bravais symbol used in these data.
• Of the 230 space-group types, 22 form 11 enantiomorphic pairs related by a change of handedness. These are a subset of the 65 Sohncke types, whose operations all preserve orientation; the other 43 Sohncke types are not members of enantiomorphic pairs. See the IUCr explanation of this distinction. The pairs are:
  • P 4₁ - (#76) and P 4₃ - (#78)
  • P 4₁ 2 2 - (#91) and P 4₃ 2 2 - (#95)
  • P 4₁ 2₁ 2 - (#92) and P 4₃ 2₁ 2 - (#96)
  • P 3₁ - (#144) and P 3₂ - (#145)
  • P 3₁ 1 2 - (#151) and P 3₂ 1 2 - (#153)
  • P 3₁ 2 1 - (#152) and P 3₂ 2 1 - (#154)
  • P 6₁ - (#169) and P 6₅ - (#170)
  • P 6₂ - (#171) and P 6₄ - (#172)
  • P 6₁ 2 2 - (#178) and P 6₅ 2 2 - (#179)
  • P 6₂ 2 2 - (#180) and P 6₄ 2 2 - (#181)
  • P 4₁ 3 2 - (#213) and P 4₃ 3 2 - (#212)