Geodesic
Some definitions of Klein's simple group of order 168 and other groups
Glasgow mathematical journal, Tome 5 (1962) no. 4, pp. 166-175

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In a previous note [3], Mennicke and I showed that the relations(8, 7|2, 3): A8=B7=(AB)2=(AB)2=(A-1B)2=(A-1B)3=Edefine a group of order 10752. As we remarked, the results of §§ 2, 3 of that note are not restricted in their application to this group; they apply to the group[3, 7]+: B7=(AB)2=(A-1B)3=Eand to any factor group of this group which in turn has Klein's simple group of order 168, defined by(4,7|2, 3): A4=B7=(AB)2=(A-1B)3=E,as a factor group. In this note I use these results to establish alternative “weaker” definitions for Klein's group and for two groups discussed by Sinkov [4], namely (8, 7|2, 3) defined above and a factor group of this group of order 1344. These latter groups are eloquently discussed by Coxeter [1]. 
Leech, John. Some definitions of Klein's simple group of order 168 and other groups. Glasgow mathematical journal, Tome 5 (1962) no. 4, pp. 166-175. doi: 10.1017/S2040618500034547
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[1] 1.Coxeter, H. S. M., The abstract group G3, 7, 15, Proc. Edinburgh Math. Soc. (2) 12 (1962). Google Scholar

[2] 2.Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for abstract groups (Ergebnisse dcr Math. N.F. 14, Springer, 1957), Chapter 2. Google Scholar | DOI

[3] 3.Leech, J. and Mennicke, J., Note on a conjecture of Coxeter, Proc. Glasgow Math. Assoc. 5, (1961), 25–29. Google Scholar | DOI

[4] 4.Sinkov, A., On the group-defining relations (2, 3, 7; p), Ann. of Math. (2) 38 (1937), 577–584. Google Scholar | DOI

[5] 5.Todd, J. A. and Coxeter, H. S. M., A practical method for enumerating cosets of a finite abstract group, Proc. Edinburgh Math. Soc. (2) 5 (1936), 26–34. Google Scholar | DOI

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