# Counting Crystallographic Groups in Low Dimensions

@article{Plesken2000CountingCG, title={Counting Crystallographic Groups in Low Dimensions}, author={Wilhelm Plesken and Tilman Schulz}, journal={Experimental Mathematics}, year={2000}, volume={9}, pages={407 - 411} }

We present the results of our computations concerning the space groups of dimension 5 and 6. We find 222 018 and 28927922 isomorph ism types of these groups, respectively. Some overall statistics on the number of Q-classes and Z-classes in dimensions up to six are provided. The computations were done with the package CARAT, which can parametrize, construct and identify all crystallographic groups up to dimension 6.

## 30 Citations

### Crystallographic point groups in five dimensions

- Mathematics
- 2010

Abstract This paper describes an approach to the deduction and labeling of crystallographic point groups in n-dimensional spaces where n is an odd number. It shows that point groups in such spaces…

### Computation of Five-and Six-Dimensional Bieberbach Groups

- MathematicsExp. Math.
- 2001

The basis of an algorithm that decides torsion-freeness of a crystallographic group as well as the triviality of its centre is described, which handles enumeration, construction, recognition and comparison problems for crystallographic groups up to dimension 6.

### Algorithms for Crystallographic Groups

- Mathematics
- 2006

This article presents a survey of the algorithms for space groups and crystallographic groups available in the computer algebra system Gap and in the software packages Carat and Cryst. © 2005 Wiley…

### Algorithm for D-V cells and fundamental domains, E 4 space groups with broken translations in the icosahedral family

- Computer Science
- 2002

This work extends its algorithm for E 4 -space groups in the icosahedral family, where non-lattice translations (broken translations) occur as well, and obtains new 4-polytopes as fundamental domains from D-V cells of crystallographic orbits.

### Crystallography in Spaces E 2 , E 3 , E 4 , E 5 ...N 0 I Isomorphism Classes: Properties and Applications to the Study of Incommensurate Phase Structures, Molecular Symmetry Groups and Crystal Families of Space E 5

- Mathematics
- 2015

This paper mainly consists of the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes. An isomorphism class is defined by a type…

### Crystallography in Spaces E 2 , E 3 , E 4 , E 5 ··· NI Isomorphism Classes: Properties and Applications to the Study of Incommensurate Phase Structures, Molecular Symmetry Groups and Crystal

- Mathematics
- 2015

This paper mainly consists of the classification of all crystallographic point groups of n-dimen- sional space with n ≤ 6 into different isomorphism classes. An isomorphism class is defined by a type…

### The four-dimensional magnetic point and space groups

- Mathematics
- 2006

Abstract This paper describes the classification of magnetic point and space groups which are also referred to as antisymmetry groups or black-and-white groups. These groups play an important role in…

### Distinguishing crystallographic groups by their finite quotients

- MathematicsJournal of Algebra
- 2021

## References

SHOWING 1-10 OF 16 REFERENCES

### Crystallographic Algorithms and Tables

- Materials Science
- 1998

A survey of definitions, theorems and algorithms for crystallographic groups are given in a dimension-independent fashion. These and some tables (including the Bravais groups up to dimension 6) form…

### Crystallographic Groups of Four-Dimensional Space

- Materials Science
- 1978

This book describes two-, three-, and four-dimensional crystallographic groups, primarily by the use of tables. Complete groups of four-dimensional tables are presented for the first time. The…

### Computation of Five-and Six-Dimensional Bieberbach Groups

- MathematicsExp. Math.
- 2001

The basis of an algorithm that decides torsion-freeness of a crystallographic group as well as the triviality of its centre is described, which handles enumeration, construction, recognition and comparison problems for crystallographic groups up to dimension 6.

### Computing Isometries of Lattices

- MathematicsJ. Symb. Comput.
- 1997

An implementation of the algorithm has been successfully applied to lattices up to dimension 40 and allows, for example, obtaining of generators for the automorphism group of the Leech lattice in less than 30 min on a HP 9000/730 workstation.

### Computing the subgroup lattice of a permutation group

- Mathematics
- 1990

In this paper, we describe a new method for computing the complete subgroup lattice of a finite group. It has been implemented within the MAGMA computational algebra system (Bosma et al., 1997) for…

### On maximal finite irreducible subgroups of (,). I. The five and seven dimensional cases

- Mathematics
- 1977

General methods for the determination of maximal finite absolutely irreducible subgroups of GL(n, Z) are described. For n = 5, 7 all these groups are computed up to Z-equivalence.

### Dual Cones and the Voronoi Algorithm

- MathematicsExp. Math.
- 2001

Max Koecher's axiomatic treatment of self-dual cones is generalized to pairs of dual cones. This leads to a powerful algorithm, à la Voronoi, to calculate the normalizer NGLn(Z)(G) and to decide…