# Difference between revisions of "M(16,2,2)"

M(16,2,2) - $k((C_4 \times C_4):C_3)$
Representative: $k((C_4 \times C_4):C_3)$ $C_4 \times C_4$ $C_3$ 8 3 1  $\left( \begin{array}{ccc} 6 & 5 & 5 \\ 5 & 6 & 5 \\ 5 & 5 & 6 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O}((C_4 \times C_4):C_3)$ $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array}\right)$ 1 $S_3$ {{{PIgroup}}} No Yes Forms its own derived equivalence class Yes M(16,2,1) {{{pcoveringblocks}}}

## Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>

Relations w.r.t. $k$: abca=bcab=cabc=0, dfed=fedf=edfe=0, ad=fc, be=da, cf=eb

## Projective indecomposable modules

Labelling the simple $B$-modules by $1,2,3$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccc} \begin{array}{c} 1 \\ 2 \ 3 \\ 3 \ 1 \ 2 \\ 1 \ 2 \ 3 \ 1 \\ 3 \ 1 \ 2 \\ 2 \ 3 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 1 \ 3 \\ 3 \ 2 \ 1 \\ 2 \ 1 \ 3 \ 2 \\ 3 \ 2 \ 1 \\ 1 \ 3 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 1 \ 2 \\ 2 \ 3 \ 1 \\ 3 \ 1 \ 2 \ 3 \\ 2 \ 3 \ 1 \\ 1 \ 2 \\ 2 \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.