Non-abelian groups of order less than or equal to 150

Question

Please is there anywhere one could see a classification of nonabelian groups of orders less than or equal to 150?

Answer 1

You could download GAP or Sage or acquire a copy of Magma. In GAP there's a function called $$\text{"AllSmallGroups($N$)"}$$ which gives you a list of all the groups of order $N$ up to isomorphism. For each one you can then use "StructureDescription(G)" for GAP to give you a short description of the group in terms of common names of groups, (semi)direct products, (non)split extensions, etc. Sage uses GAP, so it will have similar functions (with possibly different names/syntax). Magma also has this function.

By the way, there are LOTS of nonabelian groups of order $\le 150$, and most of them won't have an easy description. The description given by "StructureDescription" is also not even an isomorphism invariant - Two isomorphic groups could have different "StructureDescriptions" (for example $D_6 = \mathbb{Z}/3\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$, and two nonisomorphic groups may have the same StructureDescription. For example, even describing a simple semidirect product requires not only the two groups involved, but also a homomorphism from one into the automorphism group of the other. That's a significant amount of information - even if you only list where the homomorphism sends the generators, you still need to set up notation for how you will refer to those generators. The purpose of StructureDescription is only to give you a quick, short, and rough understanding of the group.