General description

Billiet, Y.

doi:10.1107/97809553602060000542

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 51 | 1 | 2 |

Section 2.1.5.1. General description

Y. Billiet^c

2.1.5.1. General description

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Maximal subgroups of index higher than 4 have index p, [p^2] or [p^3] , where p is prime, are necessarily isomorphic subgroups and are infinite in number. Only a few of them are listed in IT A in the block `Maximal isomorphic subgroups of lowest index IIc'. Because of their infinite number, they cannot be listed individually, but are listed in this volume as members of series under the heading `Series of maximal isomorphic subgroups'. In most of the series, the HM symbol for each isomorphic subgroup $[{\cal H} \,\lt\, {\cal G}]$ will be the same as that of $[{\cal G}]$ . However, if $[{\cal G}]$ is an enantiomorphic space group, the HM symbol of $[{\cal H}]$ will be either that of $[{\cal G}]$ or that of its enantiomorphic partner.

Example 2.1.5.1.1

Two of the four series of isomorphic subgroups of the space group [P4_1] , No. 76, are (the data on the generators are omitted):

[p]	$[{\bf c}'=p{\bf c}]$
	$[P4_3\,\,(78)]$	; $[p\equiv 3]$ (mod 4)	$[{\bf a},{\bf b},p{\bf c}]$
		no conjugate subgroups
	$[P4_1\,\,(76)]$	; $[p\equiv 1]$ (mod 4)	$[{\bf a},{\bf b},p{\bf c}]$
		no conjugate subgroups

On the other hand, the corresponding data for [P4_3] , No. 78, are

[p]	$[{\bf c}'=p{\bf c}]$
	$[P4_3\,\,(78)]$	; $[p\equiv 1]$ (mod 4)	$[{\bf a},{\bf b},p{\bf c}]$
		no conjugate subgroups
	$[P4_1\,\,(76)]$	; $[p\equiv 3]$ (mod 4)	$[{\bf a},{\bf b},p{\bf c}]$
		no conjugate subgroups

Note that in both tables the subgroups of the type [P4_3] , No. 78, are listed first because of the rules on the sequence of the subgroups.

If an isomorphic maximal subgroup of index $[i \leq 4 ]$ is a member of a series, then it is listed twice: as a member of its series and individually under the heading `Enlarged unit cell'.

Most isomorphic subgroups of index 3 are the first members of series but those of index 2 or 4 are rarely so. An example is the space group [P4_2] , No. 77, with isomorphic subgroups of index 2 (not in any series) and 3 (in a series); an exception is found in space group [P4] , No. 75, where the isomorphic subgroup for $[{\bf c}'=2{\bf c}]$ is the first member of the series $[[p]\, {\bf c}'=p{\bf c}]$ .

References

International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 51