On a correspondence between subperiodic and magnetic space groups

Question

The Wikipedia article on space groups says

Frieze groups are magnetic 1D line groups and layer groups are magnetic wallpaper groups…

Indeed:

• There are $7$ frieze groups and $7$ magnetic 1D line groups, where a magnetic group is a space group where the domains can also be time-reversed.
• There are $80$ layer groups and $80$ magnetic wallpaper groups. One day, with too much free time on my hands, I worked out the exact correspondence between these two sets of groups:

(The numbers beside each layer/magnetic wallpaper group refer to their number in the International Tables for Crystallography.)

Define an $(n,m)$ space group as a (non-magnetic) space group in $n$ dimensions whose translation lattice is $m$-dimensional. Then the two equivalences mentioned at the beginning of this question may be restated as

• There are as many $(2,1)$ space groups as magnetic $(1,1)$ space groups
• There are as many $(3,2)$ space groups as magnetic $(2,2)$ space groups

The reason for the correspondence seems clear: time reversal corresponds to reflection in a dimension orthogonal to all other dimensions. Based on this, it seemed very reasonable for me to conjecture that

• There are as many $(4,3)$ space groups as magnetic $(3,3)$ space groups

However, this is false. There are $1651$ magnetic $(3,3)$ space groups, but only $1594$ $(4,3)$ space groups. How does the discrepancy arise?

Answer 1

B. Souvignier (2006), "The four-dimensional magnetic point and space groups", Zeitschrift für Kristallographie 221, pp. 77–82 gives the number of magnetic $(3,3)$ space groups as $1594\ (+57)$ where $57$ is the number of enantiomorphic pairs.

There are no enantiomorphic pairs among the 1D line or wallpaper groups, which explains the equality in their respective dimensions, but there are $11$ enantiomorphic pairs of 3D space groups: $$P4_1,P4_3\ (6)\qquad P4_122,P4_322\ (8)\qquad P4_12_12,P4_32_12\ (8)$$ $$P3_1,P3_2\ (3)\qquad P3_112,P3_212\ (4)\qquad P3_121,P3_221\ (4)$$ $$P6_1,P6_5\ (4)\qquad P6_2,P6_4\ (4)\qquad P6_122,P6_522\ (6)\qquad P6_222,P6_422\ (6)$$ $$P4_132,P4_332\ (4)$$ When elaborated into magnetic groups, these $11$ pairs become $57$, explaining the discrepancy completely; the numbers in parentheses above indicate the number of pairs of magnetic groups corresponding to each pair under the BNS setting (verified by the Bilbao Crystallographic Server). An enantiomorphic pair of magnetic space groups collapses into one $(4,3)$ space group when time reversal is converted into reflection in the fourth dimension.