  In this paper we study symmetry properties of the Hilbert transformation of
several real variables in the Clifford algebra setting. In order to describe
the symmetry properties we introduce the group $r\mathrm{Spin}(n)+\mathbb{R}^n,
r>0,$ which is essentially an extension of the ax+b group. The study concludes
that the Hilbert transformation has certain characteristic symmetry properties
in terms of $r\mathrm{Spin}(n)+\mathbb{R}^n.$ In the present paper, for $n=2$
and $3$ we obtain, explicitly, the induced spinor representations of the
$r\mathrm{Spin}(n)+\mathbb{R}^n$ group. Then we decompose the natural
representation of $r\mathrm{Spin}(n)+\mathbb{R}^n$ into the direct sum of some
two irreducible spinor representations, by which we characterize the Hilbert
transformation in $\mathbb{R}^3$ and $\mathbb{R}^2.$ Precisely, we show that a
nontrivial skew operator is the Hilbert transformation if and only if it is
invariant under the action of the $r\mathrm{Spin}(n)+\mathbb{R}^n, n=2,3,$
group.
