u
 \kern-27pt\raise9.5pt\hbox{$\diagup$}
 \kern -3pt\raise18.5pt
\hbox{$\node$\rlap{\raise 2pt\hbox{$\kern 1pt\scriptstyle v$}}}
 \kern -2pt\raise9.5pt\hbox{$\diagdown$} & \nnode s\bar10pt\trianglerel tvu\bar10pt\nnode w & 
\nnode s\bar10pt\trianglerel tvu\bar10pt\nnode w\bar10pt\nnode x \\
\hline
\mbox{Coxeter\ element} & stu & uts & stvu & wvtsu & xwuvts \\
\hline
\end{array}$$

Using the HAP package for GAP4 we then obtained the homologies described in Table \[tableexc\] (we recall in Table \[tableexcCox\] the ones obtained earlier by Salvetti for the Coxeter groups) except for the groups $G_{12}, G_{13}, G_{22}, G_{31}$, which are not well-generated, as well as the $H_3(B,\Z)$ of type $G_{33}$. When $W$ has type $G_{13}$, the group $B$ is the same as when $W$ has Coxeter type $I_2(6)$, and the result is known. For $G_{12}$ and $G_{22}$ one can use Garside monoids introduced by M. Picantin in [@THESEPIC].

A complex for $G_{31}$ can be obtained from the theory of Garside categories by considering it as the the centralizer of some regular element in the Coxeter group $E_8$. This viewpoint was used in [@BESSISKPI1] in order to prove that the corresponding spaces $X$ and $X/W$ are $K(\pi,1)$. More precisely, a simplicial complex (reminiscent from the Charney-Meyer-Wittlesey complex) is constructed in [@BESSISKPI1], which is homotopically equivalent to $X/W$. From this construction, we got a 