  In this fith part, (with the notations of the preceding parts) we make the
following hypothesis: $(W,S)$ is a Coxeter system, irreducible, $2$-spherical
and $S$ is of cardinality $3$. Let $R:W\to GL(M)$ be a reducible reflection
representation of $W$. Let $G:= Im\,R$. Each sub-space of $M$ $(\neq M)$
stabilize by $G$ is contained in $C_{M}(G)$. Let $M':=M/C_{M}(G)$ and
$N(G):=\{g|g\in G,g\, \text{acts trivially on}\,M'\}$. We call $N(G)$ the
translation sub-group of $G$. One of the goals of this part is to study $M'$
and $N(G)$. In particular it is shown that $N(G)$ is an $\mathcal{O}G-module$.

Group Theory