# Is there a characterization of CI-groups of order less than 100?

We know some benefit criterion in articles such as:

C‎. ‎H‎. ‎Li‎, ‎On isomorphisms of finite Cayley graphs-a survey‎, ‎Discrete Math.‎, ‎256 (2002) 301-334‎.

C‎. ‎H‎. ‎Li‎, ‎Z‎. ‎P‎. ‎Lu‎, ‎P‎. ‎P‎. ‎Pa'lfy‎, ‎Further restrictions on the structure of finite CI-groups‎, ‎J‎. ‎Algebraic Combin.‎, ‎26 (2007) 161-181‎.‎

Now, do you know which finite groups of order less than 100 are CI-groups?

For example, which groups of order 30 are CI-groups?

• For completeness, a finite group is called CI if for any 2 symmetric subsets $S,T$ such that no automorphism of $G$ maps $S$ to $T$, the corresponding Cayley graphs (with no orientation, labeling, or multiple loops) are non-isomorphic. – YCor Apr 16 '16 at 16:37
• I saw a talk about this recently, and there are certainly some open cases of order less than $100$. On the other hand, it is easy to find the answer by brute-force (by computer) up to order $50$ or $60$ or so. In particular, $30$ is certainly "known". (In the sense that some people know the answer, although it might not be in the literature.) – verret Apr 16 '16 at 22:38

## 1 Answer

I know quite a lot about CI groups of small orders, but even then I don't know about every group of order up to 100, though I am gradually working through them.

For order 30, it is fairly easy to ask GAP or Magma what the answer is, and the answer is that all four groups of order 30 are CI-groups.

For each larger group, we can first test it as follows:

• if the group has any non-CI subgroups, then it is not CI
• if the group has any non-CI quotients, then it is not CI

If the group passes those two tests, then it might be CI, and so it has to be checked somehow, either using results from papers like the ones you have quoted or checking by computer.