Hi Dawn
I take issue with your statement that "according to the International
Tables for Crystallography Volume A: Space-group symmetry ..., space group
5 should be C121. It absolutely does not say that. In fact it gives 6
equally "correct" alternative settings for space group #5 (standard symbol
C2): C121 (p.126 in my edition of 2005, admittedly older than yours but I
doubt that anything of consequence has changed), then A121 and I121 are
shown on the following page. Those 3 are unique axis b settings: there
follow 3 c-unique settings A112, B112 and I112 that are used in small
molecule crystallography; the convention in MX is monoclinic b-unique.
Each page for SG #5 shows the standard symbol in the top left corner (as
explained in subsection 2.2.3 Headline in my edition):
"Note on standard monoclinic space-group symbols: In order to facilitate
recognition of a monoclinic space-group type, the familiar short symbol for
the b-axis setting (e.g. P21/c for No. 14 or C2/c for No. 15) has been
adopted as the standard symbol for a space-group type. It appears in the
headline of every description of this space group and thus does not carry
any information about the setting or the cell choice of this particular
description."
In other words C2 is not a setting symbol, simply because it carries no
information about the setting. C121, A121 and I121 are the setting symbols
(for b axis unique). It does not specify which of those is the
conventional choice, in fact they all are, depending on the convention you
choose. Note that C2 is also the standard symbol for c axis unique
A112, B112 and I112; there's no such setting as C112. As explained above
C2 is just a label; it could be anything since it carries no information.
In set theory C2 is an "equivalence class"; it's a subset of the set of all
possible settings of all the 230 space groups in which the members of the
subset (C121, A121, I121, A112, B112, I112 etc) are related by an
"equivalence relation", in this case an allowed rotation of the
co-ordinate axes ("setting").
Which of the conventional settings you choose in a particular case is
entirely up to you. The default convention that one would use in the
absence of any other information is specified in Chapter 9.1 "Special bases
for lattices" where for monoclinic SGs the convention is the cell with the
shortest a and c vectors, which comes down to beta nearest to 90 (but >=
90). Other information usually includes isomorphism with an existing
structure: if an existing isomorphous structure is in C121 you would choose
that, simply for convenience of comparisons. Equally, if there's an I121
isomorphous structure you would choose that. In the absence of any
isomorphous structure you would choose the one with beta nearest to 90.
It's quite common for an isomorphous structure of C2 to be in
orthorhombic I222 where the crystallographic 2-folds superpose on NCS
2-folds parallel to the axes in I2, so in that case it would be helpful to
choose I2.
Hope this helps!
-- Ian
On Mon, 10 Nov 2025 at 10:31, Dawn Duan <
[email protected]> wrote:
> Dear community members,
>
> I recently received a different opinion regarding whether one sample
> should be C121 or I121.
>
> I am trying to integrate automatic data processing and displaying resutls
> to users. In the pipeline, I run pointless to get the best space group
> solution, in the end of this message comes the detailed pointless output
> related to space group determination that I got for one of the samples. I
> took the best solution from pointless, i.e. I121 (Cell: 81.74 67.10
> 86.75 90.00 98.20 90.00). However, I later also extracted the
> space group and unit cell result from the phenix.xtriage output and got
> C121 (Cell: 110.377 67.1 81.744 90 128.931 90). The space group number
> indicates space group 5 from both cases, and according to the International
> Tables for Crystallography Volume A: Space-group symmetry Second online
> edition (2016) [ISBN: 978-0-470-97423-0 (
> https://it.iucr.org/services/purchase/)[doi:
> 10.1107/97809553602060000114](
> http://dx.doi.org/10.1107/97809553602060000114), space group 5 should be
> C121.
>
> Could I receive some wisdom from the community regarding whether one is
> correct, and if so, which one, or whether both are correct? Any further
> comments/opinion regarding whether I should display both options (I121 and
> C121 with their corresponding unit cell parameters) would be helpful.
>
> Thank you so much in advance!
>
> Best,
> Dawn
>
> pointless output:
> <!--SUMMARY_BEGIN-->
>
> Input file is merged so subgroups of the input symmetry are omitted
> Only groups of the same or higher symmetry are kept
> WARNING: very low completeness 0.38 in merged file, may not be able to
> determine Laue group
>
>
> Laue Group Lklhd NetZc Zc+ Zc- CC CC- Rmeas R-
> Delta ReindexOperator
>
> = 1 I 1 2/m 1 *** 1.000 8.53 8.53 0.00 0.85 0.00 0.07 0.00
> 0.0 [-l,k,h+l]
>
>
> <!--SUMMARY_END-->
>
> ********************************************************
>
> Testing Lauegroups for systematic absences
> ------------------------------------------
>
> No systematic absence zones to test in this lattice group
>
> Time for systematic absence tests: 0.053 secs
>
>
> Possible spacegroups:
> --------------------
> Indistinguishable space groups are grouped together on successive lines
> 'Reindex' is the operator to convert from the input hklin frame to the
> standard spacegroup frame.
> 'TotProb' is a total probability estimate (unnormalised)
> 'SysAbsProb' is an estimate of the probability of the space group based on
> the observed systematic absences.
> 'Conditions' are the reflection conditions (absences)
> Spacegroup TotProb SysAbsProb Reindex Conditions
>
> I 1 2 1 ( 5) 1.000 1.000 [-l,k,h+l]
>
>
> ---------------------------------------------------------------
>
>
> Space group confidence (= Sqrt(Score * (Score - NextBestScore))) = 1.00
> Laue group confidence (= Sqrt(Score * (Score - NextBestScore))) = 1.00
> Selecting space group I 1 2 1 as there is a single space group with the
> highest score
> <!--SUMMARY_BEGIN--> $TEXT:Result: $$ $$
> Best Solution: space group I 1 2 1
> Reindex operator: [-l,k,h+l]
> Laue group probability: 1.000
> Systematic absence probability: 1.000
> Total probability: 1.000
> Space group confidence: 1.000
> Laue group confidence 1.000
>
> Unit cell: 81.74 67.10 86.75 90.00 98.20 90.00
> $$ <!--SUMMARY_END-->
> HKLIN spacegroup: C 1 2 1 C-centred monoclinic
> The input crystal system is C-centred monoclinic
> (Cell: 110.38 67.10 81.74 90.00 128.93 90.00)
> The crystal system chosen for output is body-centred monoclinic
> (Cell: 81.74 67.10 86.75 90.00 98.20 90.00)
>
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