The ordinary Euclidean group? of $\mathbb{R}^n$ is the group generated from the rigid translation action of $\mathbb{R}^n$ on itself and rotations about the origin.
The super Euclidean group is analogously the supergroup of translations and rotations of the supermanifold $\mathbb{R}^{p|q}$.
Its super Lie algebra should be the super Poincare Lie algebra (up to the signature of the metric).
incomplete for the moment, to be finished off tomorrow
The following description of the super Euclidean group (once it is finished, and polished) is due to Stephan Stolz and Peter Teichner.
The data needed to define the super Euclidean group is
$V$ a $d$-dimensional inner product space
a spinor representation $\Delta^*$ of $Spin(V)$
a $Spin(V)$-equivariant map
where $Spin(V)$ is the Spin group (see Clifford algebra for the moment).
Here is the construction of $Eucl(\mathbb{R}^{d|\delta})$ for
remark $\delta$ is a multiple of $2^{[\frac{d-1}{2}]}$
set
is a complex supermanifold of dimension $(d|\delta)$
for $\delta = 1$ this is
where the last factor is $\simeq C^\infty(V; \Delta) \simeq C^\infty(V, S^+)$ where $S^+$ is the spinor bundle
now define the multiplication
by sayin what it does on sets of probes by $S$
here on the left we have the sets of sections
so we can map these as
Remark
if the data $(V, \Delta^*, \Gamma)$ and $(V', (\Delta^*)', \Gamma')$ is isomorphic we get compatible notions of structures
But if $d = 0,1,2$ and $\delta = 1$ then there is a unique such triple with non-degenerate pairing $\Gamma$ up to isomorphism.
Definition
The structure of a Euclidean supermanifold on a $(d|\delta)$-dimensional supermanifold $Y$ is a $(V \times \Pi \Delta^*, End(V, \Delta^*, \Gamma))$-structure. See there for details.
recall the Clifford algebra table:
the group structure on $V \times \Pi \Delta^*$ is that of the “translations” and “rotations”
it will be defined on generalized elements with domain $S$ by maps of sets
$\Delta^* = \mathbb{C}$
so here this is the super translation group.
$\Delta^* = \mathbb{C}$
the first map is multiplication by $u^{-1}$ and then the isomorphism on the right sends
where $z = x + i y$
translation group $V \times \Pi \Delta^* \simeq \mathbb{R}^{2|1}$
multiplication on $S$-elements
given by