A magnetic space group (MSG), also called a Shubnikov group, is a subgroup 
of the direct
product of the three-dimensional Euclidean group 
with the time-reversal group 
,
 |
whose translation subgroup is generated by three linearly independent translations. An operation carrying a prime includes time reversal, while an unprimed operation is an ordinary spatial symmetry.
Let 
be the family space group obtained by discarding
the primes from the elements of 
, and let 
be the maximal ordinary space subgroup consisting of its unprimed
elements. The magnetic space groups fall into four construct types (Shinohara et
al. 2023):
1. Type I has 
and contains no time-reversal operations. There are 230 type-I groups.
2. Type II has 
and 
,
so every spatial symmetry occurs both unprimed and primed. There are 230 type-II
groups.
3. Type III has 
,
where 
is an index-two subgroup of 
with the same translations. There are 674 type-III groups.
4. Type IV has the same coset form as type III, but 
and 
have the same point group
and different translation subgroups. There are 517 type-IV groups.
Altogether there are 1651 three-dimensional magnetic space-group types, each with a unique Belov-Neronova-Smirnova (BNS) number. A BNS number consists of two positive
integers separated by a period. The first identifies the nonmagnetic space group
for types I-III or the nonmagnetic space group of 
for type IV, while the second sequentially distinguishes magnetic
space groups associated with the same crystal family (Litvin 2014).
