# What are interesting algorithmic questions for groups in table representation?

I am currently reading about research problems in nilpotent groups ( assume table representation ). As we know that solvable group isomorphism is known to be in the (almost ) intersection of $\mathcal{NP}$ and $\text{co}\mathcal{NP}$ and even for groups whose derived series has length two, we don't how to do isomorphism in $\mathcal{NP}$ $\cap$ $\text{co}\mathcal{NP}$.

Question 1 : What are the interesting algorithmic questions to answer about nilpotent group class or its sub-class ( which is non-abelian ) other than isomorphism problem?

Question 2 : What are interesting questions to answer about groups when given in the table representation other than isomorphism? A non-interesting question is to find the intersection of two group or to find the order of any element. Testing membership is also not interesting as the input given is a table.

The following paper by Barrington-Kadau-McKenzie-Lange studies the Cayley Group Membership problem (CGM): given a group as a multiplication table, a subset of elements $X$ and an element $t$ determine if $t$ is in the subgroup generated by $X$. They show that CGM(nilpotent) is in the complexity class FO($(\log\log{n})^2$) which provably does not contain parity and is therefore not hard for any class containing parity (such as ACC$^0$, NC$^1$,L,NL etc.). This class is also not known to be contained in any of these aforementioned classes. Thus they establish that these problems are somehow orthogonal to the usual small complexity classes.