  Hopkins and Mahowald gave a simple description of the mod $p$ Eilenberg Mac
Lane spectrum $\mathrm{H}{\mathbb{F}}_p$ as the free $\mathbb{E}_2$-algebra
with an equivalence of $p$ and $0$. We show for each faithful $2$-dimensional
representation $\lambda$ of a $p$-group $G$ that the $G$-equivariant Eilenberg
Mac Lane spectrum $\mathrm{H}\underline{\mathbb{F}}_p$ is the free
$\mathbb{E}_{\lambda}$-algebra with an equivalence of $p$ and $0$. This unifies
and simplifies recent work of Behrens, Hahn, and Wilson, and extends it to
include the dihedral $2$-subgroups of $O(2)$. The main new idea is that
$\mathrm{H}\underline{\mathbb{F}}_p$ has a simple description as a $p$-cyclonic
module over $\rm{THH}(\rm{H}\mathbb{F}_p)$. We show our result is the best
possible one in that it gives all groups $G$ and representations $V$ such that
$\rm{H}\underline{\mathbb{F}}_p$ is the free $\mathbb{E}_V$-algebra with an
equivalence of $p$ and $0$.
