Counting Crystallographic Groups in Low Dimensions

Wilhelm Plesken, T. Schulz

Type: Article

Publication Date: 2000-01-01

Citations: 37

DOI: https://doi.org/10.1080/10586458.2000.10504417

Abstract

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.

Locations

Experimental Mathematics - View
Project Euclid (Cornell University) - View - PDF

Similar Works

Action	Title	Year	Authors
+	Crystallographic Algorithms and Tables	1998	Jürgen Opgenorth Wilhelm Plesken T. Schulz
+ PDF Chat	Computation of Five-and Six-Dimensional Bieberbach Groups	2001	Carlos Cid T. Schulz
+	Classification of crystallographic groups associated with Coxeter groups	1992	Dominique Martinais
+	Interactive 3D Space Group Visualization with CLUCalc and Crystallographic Subperiodic Groups in Geometric Algebra	2013	Eckhard Hitzer Christian Perwaß Daisuke Ichikawa
+	On the Fitting subgroup of almost crystallographic groups	1993	Karel Dekimpe Paul Igodt Wim Malfait
+	How many symmetry operations are needed to generate a space group?	2025	A. M. Banaru Konstantin G. Seravkin Daria A. Banaru Sergey M. Aksenov Eric A. Lord
+	C. Notations for point and crystallographic groups	2020
+	Symmetry Groups of the Plane	2014	Hayley Smith
+	Symmetry groups in the plane	2020	Lionel March Philip Steadman
+	Symmetry, Point Groups, and Character Tables.	1970	Milton Orchin H. H. Jaffé
+	D. Orbit spaces for plane crystallographic groups	2020
+	The Lower Algebraic K-Theory of Split Three-Dimensional Crystallographic Groups	2012	Daniel Farley Ivonne Johanna Ortiz
+	Character tables for point groups	2005	P. W. M. Jacobs
+	Character tables for point groups	2013	T. Wolfram Şinasi Ellialtıoğlu
+	A Matrix-algebraic Development of the Crystallographic Groups	1935	F. Seitz
+	Applications of the theory of orders to crystallographic groups	1981	Wilhelm Plesken
+	An algebraic classification of the three-dimensional crystallographic groups	1991	L. Auslander Martin L. Cook
+	A connection between different definitions of quasicrystallographic groups	1992	Sergey Piunikhin
+	Representation of Crystallographic Subperiodic Groups by Geometric Algebra	2013	Eckhard Hitzer Daisuke Ichikawa
+	Generalizations of the Tetrahedral and Octahedral Groups	1910	G. A. Miller

Works That Cite This (26)

Action	Title	Year	Authors
+	Distinguishing crystallographic groups by their finite quotients	2020	Paweł Piwek David Popović Gareth Wilkes
+	Nonexistence of locally but not globally supersymmetric orbifolds	2020	Stefan Groot Nibbelink
+ PDF Chat	Rationality Problem for Algebraic Tori	2017	Akinari Hoshi Aiichi Yamasaki
+ PDF Chat	Fibered orbifolds and crystallographic groups	2010	John G. Ratcliffe Steven T. Tschantz
+	On cohomology of crystallographic groups with cyclic holonomy of split type	2012	Nansen Petrosyan Bartosz Putrycz
+	On cohomology of crystallographic groups with cyclic holonomy of split type	2011	Nansen Petrosyan Bartosz Putrycz
+ PDF Chat	$$G_2$$-structures on flat solvmanifolds	2022	Alejandro Tolcachier
+ PDF Chat	Dual Cones and the Voronoi Algorithm	2001	Jürgen Opgenorth
+	Distinguishing crystallographic groups by their finite quotients	2019	Paweł Piwek David Popović Gareth Wilkes
+ PDF Chat	Burckhardtsche Bestimmung der Raumgruppen I	2003	Ralph Strebel

Works Cited by This (8)

Action	Title	Year	Authors
+	HANDBOOK OF MAGMA FUNCTIONS	2011	John Cannon Wieb Bosma Claus Fieker Allan Steel
+	Crystallographic Algorithms and Tables	1998	Jürgen Opgenorth Wilhelm Plesken T. Schulz
+	The lattices of six-dimensional Euclidean space	1984	Wilhelm Plesken W. Hanrath
+ PDF Chat	Dual Cones and the Voronoi Algorithm	2001	Jürgen Opgenorth
+	Computing Isometries of Lattices	1997	Wilhelm Plesken B. Souvignier
+ PDF Chat	Computation of Five-and Six-Dimensional Bieberbach Groups	2001	Carlos Cid T. Schulz
+ PDF Chat	On maximal finite irreducible subgroups of ${\rm GL}(n, {\bf Z})$. I. The five and seven dimensional cases	1977	Wilhelm Plesken Michael Pohst
+ PDF Chat	On maximal finite irreducible subgroups of 𝐺𝐿(𝑛,𝑍). II. The six dimensional case	1977	Wilhelm Plesken Michael Pohst