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Space group determination: Difference between revisions
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→Subgroup and supergroup relations of these space groups
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Compiled from [https://onlinelibrary.wiley.com/doi/book/10.1107/97809553602060000001 International Tables for Crystallography (2006) Vol. A1 (Wiley)]. Simply put, for each space group, a maximum ''translationengleiche'' subgroup has lost a single type of symmetry, and a minimum ''translationengleiche'' supergroup has gained a single symmetry type. Example: P222 is a supergroup of P2, and a subgroup of P422 (and P4222 and P23). Of course the sub-/supergroup relation is recursive, which is why P1 is also a (sub-)subgroup of P222 (but not a maximum ''translationengleiche'' subgroup). The table below does not show other types of relations, e.g. non-isomorphic ''klassengleiche'' supergroups which may result e.g. from centring translations, because I find them less relevant in space group determination. | Compiled from [https://onlinelibrary.wiley.com/doi/book/10.1107/97809553602060000001 International Tables for Crystallography (2006) Vol. A1 (Wiley)]. Simply put, for each space group, a maximum ''translationengleiche'' subgroup has lost a single type of symmetry, and a minimum ''translationengleiche'' supergroup has gained a single symmetry type. Example: P222 is a supergroup of P2, and a subgroup of P422 (and P4222 and P23). Of course the sub-/supergroup relation is recursive, which is why P1 is also a (sub-)subgroup of P222 (but not a maximum ''translationengleiche'' subgroup). The table below does not show other types of relations, e.g. non-isomorphic ''klassengleiche'' supergroups which may result e.g. from centring translations, because I find them less relevant in space group determination. | ||
The table is relevant because in particular twinning adds a symmetry type, and leads to an apparent space group which is the supergroup of the true space group. | The table is relevant because in particular (perfect) twinning adds a symmetry type, and leads to an apparent space group which is the supergroup of the true space group. | ||
{| cellpadding="0" cellspacing="0" border="1" | {| cellpadding="0" cellspacing="0" border="1" | ||
! spacegroup number | ! spacegroup number | ||