Published 29 Jul 1999. First received 19 Apr 1999.

Three-dimensional presentations for the groups of order at most 30

Graham Ellis and Irina Kholodna

Abstract: For each group G of order up to 30 we compute a small 3-dimensional CW-space X with π₁X approximately equal to G and π₂X=0, and we quantify the 'efficiency' of X. Furthermore, we give a theoretical result for treating the case when G is a semi-direct product of two groups for which 3-presentations are known. We also describe the ZG-module structure on the second homotopy group π₂X² of the 2-skeleton of X. This module structure can in principle be used to determine the cohomology groups H²(G, A) and H³(G, A) with coefficients in a ZG-module A. Our computations, which involve the Todd–Coxeter procedure for coset enumeration and the LLL algorithm for finding bases of integer lattices, are rather naive in that the LLL algorithm is applied to matrices of dimension a multiple of |G|. Thus, in their present form, our techniques can be used only on small groups (say, of order up to several hundred). They can in principle be used to construct (crossed) ZG-resolutions of Z, but again, only for small G. The paper is accompanied by two attachment files. The first of these is a summary of our computations in HTML format. The second contains various GAP programs used in the computations.

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In addition to the paper, the following electronic appendices are available to subscribers :

Appendix A : This appendix contains the updated computational results referred to in the paper Three-dimensional presentations for the groups of order at most 30, as published on the 11 June 2001. It contains a table of results, a help file on using the table, and a summary of what is contained in the entries in the table.
The original version of this appendix, published on the 29 July 1999, is available at http://www.lms.ac.uk/jcm/2/lms1999-006/appendix-a1/,

Appendix A1 : This appendix contains the computational results referred to in the paper Three-dimensional presentations for the groups of order at most 30, as originally published on the 29 July 1999. It contains a table of results, a help file on using the table, and a summary of what is contained in the entries in the table.
An updated version of this appendix is available at http://www.lms.ac.uk/jcm/2/lms1999-006/appendix-a/,

Appendix B : This appendix contains the GAP programs referred to in the paper Three-dimensional presentations for the groups of order at most 30.

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