9.5 The Macdonald Groups $G(3,m)$

Unfortunately, the groups obtained for $n=3$ already seem to be much harder. While the results for $G(3,3)$ are still very good compared to Felsch or HLT style methods it was not yet possible to compute $G(3,4)$. Here, a tuned version has to be used which uses less space. However, the results for $n=3, m=3$ show some similarities to the results obtained for $n = 2$. The strategies NONE and I-R perform best for most orderings used followed by I-ALL which is even the best for the syllable ordering syl-l-BbAa. But I-ALL is considerably worse for the length-lexicographical orderings. All other strategies perform even worse producing the best results for the syllable orderings. The best combination was I-ALL/syl-l-BAab which was not used for $n = 2$. Here, further investigations are necessary.

It should nevertheless be noted that $G(3, 3), G(3, -1)$, and $G(-1,-1)$ are all isomorphic to the generalized quaternion group of order 16 and that the coset enumeration for the latter two is almost trivial!