The lattices of six-dimensional Euclidean space

Authors:
W. Plesken and W. Hanrath

Journal:
Math. Comp. **43** (1984), 573-587

MSC:
Primary 11H06; Secondary 20H15, 52A43

DOI:
https://doi.org/10.1090/S0025-5718-1984-0758205-5

MathSciNet review:
758205

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Abstract: The lattices of full rank of the six-dimensional Euclidean space are classified according to their automorphism groups (Bravais classification). We find 826 types of such lattices.

**[1]**H. Brown, R. Bülow, J. Neubüser, H. Wondratschek & H. Zassenhaus,*Crystallographic Groups of Four-Dimensional Space*, Wiley-Interscience, New York, 1978.**[2]**H. Brown, J. Neubüser & H. Zassenhaus, "On integral groups II; The irreducible case,"*Numer. Math.*, v. 20, 1972, pp. 22-31. MR**0318335 (47:6882)****[3]**J. Cannon,*A General Purpose Group Theory Program*, Lecture Notes in Math., Vol. 372, Springer-Verlag, Berlin and New York, 1973. MR**0354823 (50:7300)****[4]**J. D. Jarrat,*The Decomposition of Crystal Families*, Report Series No. 151, Dept. of Math., Univ. of Auckland, New Zealand, 1979. MR**578269 (81i:20057)****[5]**J. Neubüser,*Computing Moderately Large Groups: Some Methods and Applications*, SIAM-AMS-Proc., Vol. 4, Amer. Math. Soc., Providence, R. I., 1971. MR**0379633 (52:538)****[6]**J. Neubüser, W. Plesken & H. Wondratschek,*An Emendatory Discursion on Defining Crystal Systems*, Proc. Conf. on Crystallographic Groups (Univ. Bielefeld, Bielefeld, 1979),*Match No.*10, 1981, pp. 77-96. MR**620802 (82i:20059)****[7]**W. Plesken,*Beiträge zur Bestimmung der endlichen irreduziblen Untergruppen von*, Dissertation, RWTH Aachen, 1974.**[8]**W. Plesken, "The Bravais group and the normalizer of a reducible finite subgroup of ,"*Comm. Algebra*, v. 5, 1977, pp. 375-396. MR**0439948 (55:12829)****[9]**W. Plesken, "On reducible and decomposable representations of orders,"*J. Reine Angew. Math.*, v. 297, 1978, pp. 188-210. MR**0466193 (57:6073)****[10]**W. Plesken,*Bravais Groups in Low Dimensions*, Proc. Conf. on Crystallographic Groups (Univ. Bielefeld, Bielefeld, 1979), Part II,*Match No.*10, 1981, pp. 97-119. MR**620803 (82k:20077)****[11]**W. Plesken, "Applications of the theory of orders to crystallographic groups," in*Integral Representations and Applications*(Proceedings, Oberwolfach, Germany 1980); edited by K. W. Roggenkamp, Lecture Notes in Math., Vol. 882, Springer-Verlag, Berlin and New York, 1981, pp. 37-92. MR**646093 (83i:82030)****[12]**W. Plesken, "Counting with groups and rings,"*J. Reine Angew. Math.*, v. 334, 1982, pp. 40-68. MR**667449 (84a:20008)****[13]**W. Plesken & M. Pohst, "On maximal finite irreducible subgroups of . I. The five and seven dimensional case,"*Math. Comp.*, v. 31, 1977, pp. 536-551; "II. The six dimensional case,"*ibid.*, pp. 552-573. MR**0444789 (56:3137a)****[14]**W. Plesken & M. Pohst, "On maximal finite irreducible subgroups of . III. The nine dimensional case,"*Math. Comp.*, v. 34, 1980, pp. 245-258; "IV. Remarks on even dimensions with applications to ,"*ibid.*, pp. 259-275; "V. The eight dimensional case and a complete description of dimensions less than ten,"*ibid.*, pp. 277-301. MR**551303 (81b:20012a)****[15]**S. S. Ryskow & S. D. Lomakina, "The proof of the theorems on the maximal finite groups of integral -matrices,"*Trudy Mat. Inst. Steklov.*, v. 152, 1980, pp. 204-215. (Russian) MR**603825 (82e:20058)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1984-0758205-5

Keywords:
Bravais lattices,
integral matrix groups

Article copyright:
© Copyright 1984
American Mathematical Society