I know that there is one group only of order 81 which is non abelian and of exponent 3. But I have no idea how to prove it.
This question is personal. I looked at Wims, which suggested the answer. The group is the direct product $G=T\times C_3$, where $T$ is the upper unipotent triangular matrices with coefficients in $\mathbb F_3$ and where $C_3$ is the cyclic group of order $3$.
In order to reach to $G$, I tried of course the standard methods, from the derived subgroup to the Frattini of such a $3$-group, which appeared all to be useless. I think now that I miss some technique which is relevant to the problem. I need the magic word behind this exercice. I will manage for the rest.
May I have any hint in this direction?
Thanks
Yvette