# M(16,2,2)

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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  $\left( \begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O}A_4$ $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ \end{array}\right)$ 1 $\mathcal{T}(B)=S_3$ {{{PIgroup}}} Yes $\mathcal{O}A_4$ Yes M(4,2,2) Yes {{{coveringblocks}}} {{{coveredblocks}}} {{{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. $k$: ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb

## Other notatable representatives

Block number 2 of $k PSL_3(7)$ in the labelling used in [1]

## Covering blocks and covered blocks

Let $N \triangleleft G$ with $p'$-index and let $B$ be a block of $\mathcal{O} G$ covering a block $b$ of $\mathcal{O} N$.

If $b$ lies in M(4,2,3), then $B$ must lie in M(4,2,1) or M(4,2,3). For example consider blocks of $PSL_3(7) \triangleleft PGL_3(7)$.

If $B$ lies in M(4,2,3), then $b$ must lie in M(4,2,1) or M(4,2,3). For example consider the principal blocks of $O_2(A_4) \triangleleft A_4$.

## Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, S_2, S_3$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccc} \begin{array}{ccc} & S_1 & \\ S_2 & & S_3 \\ & S_1 & \\ \end{array}, & \begin{array}{ccc} & S_2 & \\ S_1 & & S_3 \\ & S_2 & \\ \end{array}, & \begin{array}{ccc} & S_3 & \\ S_1 & & S_2 \\ & S_3 & \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.