Algebraic group interpretations of dihedral group:D8

Description
The group dihedral group:D8 can be interpreted as the group of $$\mathbb{F}_2$$-points of a unipotent algebraic group over the algebraic closure of $$\mathbb{F}_2$$, namely the unitriangular matrix group of degree three $$UT(3,\overline{\mathbb{F}_2})$$. Explicitly, $$D_8 \cong UT(3,\mathbb{F}_2)$$, which in turn is the set of $$\mathbb{F}_2$$-fixed points of $$\overline{\mathbb{F}_2}$$.

This is the only possible structure for $$D_8$$ as the $$\mathbb{F}_2$$-fixed points of a unipotent algebraic group.

In fact, since the group scheme corresponding to the unitriangular matrix group of degree three is integral (i.e., makes sense over $$\mathbb{Z}$$), analogues of the group can be defined over any ring.