Origin shift
Billiet, Y.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 2.1.5.3,
p.

[ doi:10.1107/97809553602060000797 ]
Example

**2.1.5.3.1**
The space group,, No. 112, has two series of maximal isomorphic subgroups . For one of them the lattice relations are, listed as . The index is . For each value of p there exist exactly conjugate subgroups ...

Generators
Billiet, Y.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 2.1.5.4,
p.

[ doi:10.1107/97809553602060000797 ]
and r, if relevant) and u (and v and w, if relevant).
Example

**2.1.5.4.1**
Space group, No. 198. In the series defined by the lattice relations and the origin shift there exist exactly conjugate subgroups for each ...

Basis transformation
Billiet, Y.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 2.1.5.2,
p.

[ doi:10.1107/97809553602060000797 ]
sufficient as a parameter.
Example

**2.1.5.2.1**
The isomorphic subgroups of the space group, No. 93, can be described by two series with the bases of their members:
.
In other cases ...

Monoclinic space groups
Billiet, Y.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 2.1.5.5.1,
p.

[ doi:10.1107/97809553602060000797 ]
...

Special series
Billiet, Y.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 2.1.5.5,
p.

[ doi:10.1107/97809553602060000797 ]
5.1.3.6
(a) and (c) in IT A.
Applying equations (

**2.1.5.3**), (

**2.1.5.1**) and (

**2.1.5.2**), one gets From equation (

**2.1.5.4**) it follows that One obtains Y from equation (

**2.1.5.5**) by matrix multiplication ...

Series of maximal isomorphic subgroups
Billiet, Y.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 2.1.5,
p.

[ doi:10.1107/97809553602060000797 ]
to rhombohedral axes is described by where the matrices are listed in IT A, Table 5.1.3.1, see also Figs. 5.1.3.6
(a) and (c) in IT A.
Applying equations (

**2.1.5.3**), (

**2.1.5.1**) and (

**2.1.5.2**), one gets From ...

General description
Billiet, Y.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 2.1.5.1,
p.

[ doi:10.1107/97809553602060000797 ]
of the series, the HM symbol for each isomorphic subgroup will be the same as that of . However, if is an enantiomorphic space group, the HM symbol of will be either that of or that of its enantiomorphic partner.
Example

**2.1.5.1.1** ...

Trigonal space groups with rhombohedral lattice
Billiet, Y.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 2.1.5.5.2,
p.

[ doi:10.1107/97809553602060000797 ]
to rhombohedral axes is described by where the matrices are listed in IT A, Table 5.1.3.1, see also Figs. 5.1.3.6
(a) and (c) in IT A.
Applying equations (

**2.1.5.3**), (

**2.1.5.1**) and (

**2.1.5.2**), one gets From ...

Space groups with two origin choices
Billiet, Y.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 2.1.5.5.3,
p.

[ doi:10.1107/97809553602060000797 ]
of the two series related by the origin shift is similar; there are only differences in the generators.
Example

**2.1.5.5.2**
Consider the space group, No. 48, in both origin choices and the corresponding series defined by and . In origin ...