Isomorphic groups algebras with non-isomorphic groups There is an interesting result of Dade about isomorphism of group algebras: > There exists finite non-isomorphic metabelian groups $G$ and $H$ whose group algebras over any field are isomorphic. I was looking for the examples of these groups, but I didn't find them; only the result is stated in most of the books or slides online. So, I have one obvious question and one modification of it. **Question 1.** What are the (counterexamples) groups constructed by Dade? **Question 2.** Is the example by Dade, the smallest order example?

You can find Dade's original paper at <

He gives a construction of pairs of groups of order $p^3q^3$ for arbitrary primes $p,q$ with $q\equiv 1\pmod{p^2}$, so his smallest examples would be of order $1000$ ($p=2$ and $q=5$).

I've no idea whether smaller examples are possible using a different construction.