We construct a non-abelian extension $\Gamma$ of $S^1$ by $\cy 3 \times \cy 3$, and prove that $\Gamma$ acts freely and smoothly on $S^{ 5} \times S^{ 5}$. This gives new actions on $S^{ 5} \times S^{ 5}$ for an infinite family $\cP$ of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group $G_2$ admits a free smooth action on $S^{ 11}\times S^{ 11}$. This gives new actions on $S^{ 11}\times S^{ 11}$ for an infinite family $\cE $ of finite groups. We explain the significance of these families $\cP $, $\cE $ for the general existence problem, and correct some mistakes in the literature.
Categories: math.AT math.GT