Difference between revisions of "M(32,51,16)"
(Created page with "{{blockbox |title = M(32,51,16) - <math>B_0(k(SL_2(8) \times A_5))</math> |image = |representative = <math>B_0(k(SL_2(8) \times A_5))</math> |defect = (C2)%5E5|<ma...") |
(→Covering blocks and covered blocks) |
||
| (One intermediate revision by the same user not shown) | |||
| Line 37: | Line 37: | ||
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>. | Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>. | ||
| − | If <math>b</math> is in M(32,51,16), then <math>B</math> is in M(32,51,16) or [[M(32,51,29]]. | + | If <math>b</math> is in M(32,51,16), then <math>B</math> is in M(32,51,16) or [[M(32,51,29)]]. |
== Projective indecomposable modules == | == Projective indecomposable modules == | ||
| Line 43: | Line 43: | ||
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{21}</math>, the projective indecomposable modules have Loewy structure as follows: | Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{21}</math>, the projective indecomposable modules have Loewy structure as follows: | ||
| − | <math> | + | <math> |
\begin{array}{c} | \begin{array}{c} | ||
S_{1} \\ | S_{1} \\ | ||
| Line 56: | Line 56: | ||
S_{4} S_{5} S_{2} S_{6} S_{3} \\ | S_{4} S_{5} S_{2} S_{6} S_{3} \\ | ||
S_{1} \\ | S_{1} \\ | ||
| − | \end{array} | + | \end{array}</math> <math> |
| − | |||
\begin{array}{c} | \begin{array}{c} | ||
S_{2} \\ | S_{2} \\ | ||
| Line 71: | Line 70: | ||
S_{2} \\ | S_{2} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
| − | |||
| − | |||
| − | |||
| − | <math> | ||
\begin{array}{c} | \begin{array}{c} | ||
S_{3} \\ | S_{3} \\ | ||
| Line 89: | Line 84: | ||
S_{3} \\ | S_{3} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{4} \\ | S_{4} \\ | ||
| Line 103: | Line 98: | ||
S_{4} \\ | S_{4} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
| − | |||
| − | |||
| − | |||
| − | <math> | ||
| − | |||
\begin{array}{c} | \begin{array}{c} | ||
S_{5} \\ | S_{5} \\ | ||
| Line 122: | Line 112: | ||
S_{5} \\ | S_{5} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{6} \\ | S_{6} \\ | ||
| Line 136: | Line 126: | ||
S_{6} \\ | S_{6} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{7} \\ | S_{7} \\ | ||
| Line 150: | Line 140: | ||
S_{7} \\ | S_{7} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{8} \\ | S_{8} \\ | ||
| Line 164: | Line 154: | ||
S_{8} \\ | S_{8} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
| − | </math> | ||
| − | |||
| − | |||
| − | |||
| − | <math> | ||
| − | |||
\begin{array}{c} | \begin{array}{c} | ||
S_{9} \\ | S_{9} \\ | ||
| Line 184: | Line 168: | ||
S_{9} \\ | S_{9} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{10} \\ | S_{10} \\ | ||
| Line 198: | Line 182: | ||
S_{10} \\ | S_{10} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{11} \\ | S_{11} \\ | ||
| Line 212: | Line 196: | ||
S_{11} \\ | S_{11} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{12} \\ | S_{12} \\ | ||
| Line 226: | Line 210: | ||
S_{12} \\ | S_{12} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
| − | </math> | ||
| − | |||
| − | |||
| − | |||
| − | <math> | ||
| − | |||
\begin{array}{c} | \begin{array}{c} | ||
S_{13} \\ | S_{13} \\ | ||
| Line 246: | Line 224: | ||
S_{13} \\ | S_{13} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{14} \\ | S_{14} \\ | ||
| Line 260: | Line 238: | ||
S_{14} \\ | S_{14} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{15} \\ | S_{15} \\ | ||
| Line 274: | Line 252: | ||
S_{15} \\ | S_{15} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{16} \\ | S_{16} \\ | ||
| Line 288: | Line 266: | ||
S_{16} \\ | S_{16} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
| − | </math> | ||
| − | |||
| − | |||
| − | |||
| − | <math> | ||
| − | |||
\begin{array}{c} | \begin{array}{c} | ||
S_{17} \\ | S_{17} \\ | ||
| Line 308: | Line 280: | ||
S_{17} \\ | S_{17} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{18} \\ | S_{18} \\ | ||
| Line 322: | Line 294: | ||
S_{18} \\ | S_{18} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{19} \\ | S_{19} \\ | ||
| Line 336: | Line 308: | ||
S_{19} \\ | S_{19} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{20} \\ | S_{20} \\ | ||
| Line 350: | Line 322: | ||
S_{20} \\ | S_{20} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> <math> | |
\begin{array}{c} | \begin{array}{c} | ||
S_{21} \\ | S_{21} \\ | ||
| Line 364: | Line 336: | ||
S_{21} \\ | S_{21} \\ | ||
\end{array} | \end{array} | ||
| − | + | </math> | |
| − | </math> | ||
== Irreducible characters == | == Irreducible characters == | ||
Latest revision as of 17:16, 8 December 2019
| Representative: | [math]B_0(k(SL_2(8) \times A_5))[/math] |
|---|---|
| Defect groups: | [math](C_2)^5[/math] |
| Inertial quotients: | [math]C_{21}[/math] |
| [math]k(B)=[/math] | 32 |
| [math]l(B)=[/math] | 21 |
| [math]{\rm mf}_k(B)=[/math] | 1 |
| [math]{\rm Pic}_k(B)=[/math] | |
| Cartan matrix: | See below. |
| Defect group Morita invariant? | Yes |
| Inertial quotient Morita invariant? | Yes |
| [math]\mathcal{O}[/math]-Morita classes known? | Yes |
| [math]\mathcal{O}[/math]-Morita classes: | [math]B_0(\mathcal{O} (SL_2(8) \times A_5))[/math] |
| Decomposition matrices: | See below. |
| [math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |
| [math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | |
| [math]PI(B)=[/math] | |
| Source algebras known? | No |
| Source algebra reps: | |
| [math]k[/math]-derived equiv. classes known? | Yes |
| [math]k[/math]-derived equivalent to: | M(32,51,13), M(32,51,14), M(32,51,15) |
| [math]\mathcal{O}[/math]-derived equiv. classes known? | Yes |
| [math]p'[/math]-index covering blocks: | |
| [math]p'[/math]-index covered blocks: | |
| Index [math]p[/math] covering blocks: |
Contents
Basic algebra
Other notatable representatives
Covering blocks and covered blocks
Let [math]N \triangleleft G[/math] with prime [math]p'[/math]-index and let [math]B[/math] be a block of [math]\mathcal{O} G[/math] covering a block [math]b[/math] of [math]\mathcal{O} N[/math].
If [math]b[/math] is in M(32,51,16), then [math]B[/math] is in M(32,51,16) or M(32,51,29).
Projective indecomposable modules
Labelling the simple [math]B[/math]-modules by [math]S_1, \dots, S_{21}[/math], the projective indecomposable modules have Loewy structure as follows:
[math] \begin{array}{c} S_{1} \\ S_{5} S_{3} S_{6} S_{4} S_{2} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{11} S_{14} S_{9} S_{13} S_{16} S_{15} S_{8} S_{10} S_{12} \\ S_{2} S_{4} S_{5} S_{5} S_{3} S_{4} S_{2} S_{4} S_{4} S_{6} S_{3} S_{2} S_{3} S_{2} S_{5} S_{5} S_{6} S_{3} S_{6} S_{6} S_{19} S_{7} S_{20} S_{18} S_{21} S_{17} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{11} S_{15} S_{9} S_{11} S_{12} S_{16} S_{8} S_{15} S_{12} S_{13} S_{10} S_{14} S_{15} S_{14} S_{16} S_{9} S_{9} S_{13} S_{13} S_{10} S_{12} S_{8} S_{8} S_{11} S_{14} S_{10} S_{16} \\ S_{6} S_{6} S_{4} S_{3} S_{3} S_{6} S_{5} S_{3} S_{6} S_{3} S_{3} S_{6} S_{6} S_{4} S_{2} S_{2} S_{4} S_{4} S_{4} S_{5} S_{5} S_{3} S_{5} S_{2} S_{2} S_{2} S_{2} S_{5} S_{5} S_{4} S_{7} S_{19} S_{7} S_{17} S_{18} S_{17} S_{20} S_{20} S_{21} S_{18} S_{19} S_{21} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{14} S_{15} S_{9} S_{10} S_{13} S_{13} S_{12} S_{8} S_{15} S_{13} S_{16} S_{10} S_{10} S_{16} S_{15} S_{8} S_{8} S_{12} S_{11} S_{9} S_{11} S_{14} S_{16} S_{9} S_{11} S_{12} S_{14} \\ S_{4} S_{2} S_{6} S_{5} S_{6} S_{4} S_{6} S_{2} S_{3} S_{3} S_{2} S_{6} S_{2} S_{5} S_{3} S_{4} S_{3} S_{5} S_{4} S_{5} S_{7} S_{18} S_{20} S_{19} S_{21} S_{17} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{13} S_{9} S_{10} S_{8} S_{11} S_{12} S_{14} S_{15} S_{16} \\ S_{4} S_{5} S_{2} S_{6} S_{3} \\ S_{1} \\ \end{array}[/math] [math] \begin{array}{c} S_{2} \\ S_{1} S_{12} S_{16} S_{14} \\ S_{2} S_{4} S_{3} S_{5} S_{2} S_{2} S_{6} S_{20} S_{18} S_{7} \\ S_{1} S_{1} S_{1} S_{1} S_{8} S_{12} S_{14} S_{16} S_{15} S_{10} S_{12} S_{14} S_{13} S_{9} S_{16} S_{11} \\ S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} S_{6} S_{6} S_{2} S_{2} S_{5} S_{2} S_{3} S_{6} S_{3} S_{2} S_{21} S_{18} S_{19} S_{7} S_{17} S_{20} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{13} S_{15} S_{9} S_{10} S_{8} S_{12} S_{15} S_{8} S_{9} S_{14} S_{12} S_{16} S_{16} S_{10} S_{13} S_{11} S_{11} S_{14} \\ S_{5} S_{6} S_{5} S_{2} S_{6} S_{2} S_{5} S_{4} S_{2} S_{4} S_{3} S_{3} S_{6} S_{3} S_{2} S_{4} S_{18} S_{21} S_{17} S_{20} S_{19} S_{7} \\ S_{1} S_{1} S_{1} S_{1} S_{10} S_{8} S_{14} S_{15} S_{14} S_{11} S_{13} S_{16} S_{9} S_{16} S_{12} S_{12} \\ S_{2} S_{6} S_{3} S_{5} S_{2} S_{2} S_{4} S_{20} S_{7} S_{18} \\ S_{1} S_{12} S_{14} S_{16} \\ S_{2} \\ \end{array} [/math] [math] \begin{array}{c} S_{3} \\ S_{1} S_{11} S_{15} S_{13} \\ S_{4} S_{3} S_{2} S_{5} S_{6} S_{3} S_{3} S_{19} S_{17} S_{21} \\ S_{1} S_{1} S_{1} S_{1} S_{16} S_{15} S_{14} S_{9} S_{11} S_{10} S_{12} S_{15} S_{11} S_{8} S_{13} S_{13} \\ S_{3} S_{6} S_{2} S_{5} S_{3} S_{5} S_{5} S_{2} S_{6} S_{3} S_{2} S_{4} S_{4} S_{6} S_{3} S_{4} S_{19} S_{18} S_{17} S_{7} S_{20} S_{21} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{10} S_{9} S_{10} S_{13} S_{13} S_{16} S_{11} S_{8} S_{12} S_{8} S_{12} S_{16} S_{15} S_{15} S_{11} S_{9} S_{14} S_{14} \\ S_{5} S_{6} S_{2} S_{5} S_{5} S_{3} S_{3} S_{3} S_{6} S_{2} S_{2} S_{4} S_{6} S_{4} S_{4} S_{3} S_{19} S_{21} S_{20} S_{17} S_{7} S_{18} \\ S_{1} S_{1} S_{1} S_{1} S_{9} S_{10} S_{11} S_{16} S_{14} S_{8} S_{15} S_{11} S_{13} S_{12} S_{15} S_{13} \\ S_{4} S_{2} S_{3} S_{5} S_{6} S_{3} S_{3} S_{17} S_{19} S_{21} \\ S_{1} S_{11} S_{15} S_{13} \\ S_{3} \\ \end{array} [/math] [math] \begin{array}{c} S_{4} \\ S_{1} S_{14} S_{8} S_{13} \\ S_{2} S_{4} S_{5} S_{6} S_{4} S_{3} S_{4} S_{7} S_{19} \\ S_{1} S_{1} S_{1} S_{1} S_{14} S_{12} S_{15} S_{11} S_{8} S_{16} S_{8} S_{13} S_{10} S_{14} S_{13} \\ S_{2} S_{2} S_{6} S_{6} S_{2} S_{4} S_{3} S_{6} S_{5} S_{3} S_{4} S_{5} S_{4} S_{3} S_{4} S_{5} S_{19} S_{7} S_{21} S_{18} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{14} S_{10} S_{13} S_{10} S_{15} S_{12} S_{13} S_{15} S_{11} S_{8} S_{8} S_{16} S_{16} S_{11} S_{12} S_{14} \\ S_{4} S_{4} S_{4} S_{2} S_{2} S_{3} S_{5} S_{3} S_{6} S_{4} S_{2} S_{5} S_{6} S_{5} S_{3} S_{6} S_{7} S_{21} S_{19} S_{18} \\ S_{1} S_{1} S_{1} S_{1} S_{12} S_{14} S_{15} S_{14} S_{16} S_{10} S_{8} S_{13} S_{13} S_{8} S_{11} \\ S_{4} S_{5} S_{4} S_{2} S_{3} S_{6} S_{4} S_{19} S_{7} \\ S_{1} S_{13} S_{14} S_{8} \\ S_{4} \\ \end{array} [/math] [math] \begin{array}{c} S_{5} \\ S_{1} S_{15} S_{9} S_{12} \\ S_{2} S_{6} S_{5} S_{5} S_{4} S_{5} S_{3} S_{20} S_{17} \\ S_{1} S_{1} S_{1} S_{1} S_{15} S_{13} S_{15} S_{9} S_{14} S_{9} S_{12} S_{8} S_{16} S_{12} S_{11} \\ S_{2} S_{2} S_{6} S_{6} S_{6} S_{3} S_{3} S_{5} S_{4} S_{5} S_{2} S_{4} S_{3} S_{5} S_{4} S_{5} S_{7} S_{17} S_{19} S_{20} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{12} S_{12} S_{14} S_{11} S_{13} S_{16} S_{14} S_{15} S_{8} S_{11} S_{15} S_{16} S_{9} S_{9} S_{13} S_{8} \\ S_{4} S_{2} S_{4} S_{5} S_{5} S_{4} S_{2} S_{2} S_{6} S_{5} S_{3} S_{3} S_{6} S_{5} S_{6} S_{3} S_{17} S_{20} S_{19} S_{7} \\ S_{1} S_{1} S_{1} S_{1} S_{13} S_{16} S_{15} S_{9} S_{14} S_{11} S_{12} S_{12} S_{9} S_{8} S_{15} \\ S_{5} S_{5} S_{2} S_{4} S_{5} S_{3} S_{6} S_{17} S_{20} \\ S_{1} S_{15} S_{12} S_{9} \\ S_{5} \\ \end{array} [/math] [math] \begin{array}{c} S_{6} \\ S_{1} S_{11} S_{10} S_{16} \\ S_{6} S_{6} S_{5} S_{4} S_{6} S_{2} S_{3} S_{18} S_{21} \\ S_{1} S_{1} S_{1} S_{1} S_{9} S_{13} S_{11} S_{16} S_{11} S_{16} S_{12} S_{10} S_{15} S_{14} S_{10} \\ S_{4} S_{4} S_{6} S_{5} S_{2} S_{6} S_{4} S_{5} S_{3} S_{6} S_{6} S_{2} S_{2} S_{3} S_{5} S_{3} S_{18} S_{20} S_{21} S_{17} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{16} S_{14} S_{13} S_{16} S_{10} S_{9} S_{10} S_{14} S_{12} S_{15} S_{13} S_{15} S_{9} S_{12} S_{11} S_{11} \\ S_{4} S_{2} S_{6} S_{6} S_{3} S_{2} S_{4} S_{3} S_{4} S_{5} S_{6} S_{5} S_{5} S_{6} S_{2} S_{3} S_{21} S_{20} S_{17} S_{18} \\ S_{1} S_{1} S_{1} S_{1} S_{12} S_{16} S_{16} S_{14} S_{15} S_{11} S_{13} S_{11} S_{9} S_{10} S_{10} \\ S_{5} S_{6} S_{6} S_{2} S_{3} S_{6} S_{4} S_{21} S_{18} \\ S_{1} S_{11} S_{16} S_{10} \\ S_{6} \\ \end{array} [/math] [math] \begin{array}{c} S_{7} \\ S_{8} S_{14} \\ S_{2} S_{4} S_{19} \\ S_{1} S_{13} S_{8} S_{12} \\ S_{2} S_{3} S_{4} S_{5} S_{7} \\ S_{1} S_{1} S_{15} S_{14} S_{14} \\ S_{5} S_{3} S_{4} S_{2} S_{7} \\ S_{1} S_{12} S_{8} S_{13} \\ S_{4} S_{2} S_{19} \\ S_{14} S_{8} \\ S_{7} \\ \end{array} [/math] [math] \begin{array}{c} S_{8} \\ S_{4} S_{19} S_{7} \\ S_{1} S_{8} S_{13} S_{8} S_{14} \\ S_{3} S_{4} S_{4} S_{2} S_{5} S_{7} S_{19} \\ S_{1} S_{1} S_{1} S_{14} S_{15} S_{8} S_{13} S_{12} \\ S_{5} S_{5} S_{4} S_{2} S_{4} S_{3} S_{2} S_{3} \\ S_{1} S_{1} S_{1} S_{14} S_{12} S_{13} S_{15} S_{8} \\ S_{4} S_{3} S_{2} S_{5} S_{4} S_{19} S_{7} \\ S_{1} S_{14} S_{13} S_{8} S_{8} \\ S_{4} S_{7} S_{19} \\ S_{8} \\ \end{array} [/math] [math] \begin{array}{c} S_{9} \\ S_{5} S_{17} S_{20} \\ S_{1} S_{12} S_{15} S_{9} S_{9} \\ S_{5} S_{5} S_{3} S_{2} S_{6} S_{20} S_{17} \\ S_{1} S_{1} S_{1} S_{12} S_{11} S_{9} S_{15} S_{16} \\ S_{3} S_{5} S_{2} S_{5} S_{3} S_{6} S_{6} S_{2} \\ S_{1} S_{1} S_{1} S_{15} S_{12} S_{16} S_{9} S_{11} \\ S_{5} S_{2} S_{6} S_{3} S_{5} S_{17} S_{20} \\ S_{1} S_{9} S_{9} S_{15} S_{12} \\ S_{5} S_{17} S_{20} \\ S_{9} \\ \end{array} [/math] [math] \begin{array}{c} S_{10} \\ S_{6} S_{18} S_{21} \\ S_{1} S_{10} S_{10} S_{16} S_{11} \\ S_{3} S_{2} S_{4} S_{6} S_{6} S_{18} S_{21} \\ S_{1} S_{1} S_{1} S_{10} S_{13} S_{11} S_{16} S_{14} \\ S_{2} S_{3} S_{3} S_{2} S_{4} S_{4} S_{6} S_{6} \\ S_{1} S_{1} S_{1} S_{11} S_{13} S_{14} S_{10} S_{16} \\ S_{3} S_{4} S_{6} S_{2} S_{6} S_{18} S_{21} \\ S_{1} S_{10} S_{16} S_{11} S_{10} \\ S_{6} S_{18} S_{21} \\ S_{10} \\ \end{array} [/math] [math] \begin{array}{c} S_{11} \\ S_{3} S_{6} S_{21} \\ S_{1} S_{11} S_{15} S_{13} S_{10} S_{16} \\ S_{5} S_{3} S_{2} S_{4} S_{6} S_{6} S_{3} S_{18} S_{17} \\ S_{1} S_{1} S_{1} S_{9} S_{14} S_{11} S_{15} S_{12} S_{16} S_{13} S_{10} S_{11} \\ S_{4} S_{3} S_{6} S_{5} S_{3} S_{5} S_{2} S_{6} S_{4} S_{2} S_{20} S_{21} S_{21} \\ S_{1} S_{1} S_{1} S_{11} S_{16} S_{10} S_{14} S_{12} S_{9} S_{13} S_{11} S_{15} \\ S_{6} S_{2} S_{6} S_{5} S_{4} S_{3} S_{3} S_{18} S_{17} \\ S_{1} S_{16} S_{15} S_{10} S_{11} S_{13} \\ S_{6} S_{3} S_{21} \\ S_{11} \\ \end{array} [/math] [math] \begin{array}{c} S_{12} \\ S_{2} S_{5} S_{20} \\ S_{1} S_{14} S_{16} S_{15} S_{12} S_{9} \\ S_{5} S_{2} S_{6} S_{2} S_{4} S_{3} S_{5} S_{17} S_{7} \\ S_{1} S_{1} S_{1} S_{15} S_{12} S_{16} S_{11} S_{8} S_{13} S_{14} S_{9} S_{12} \\ S_{4} S_{4} S_{2} S_{6} S_{5} S_{3} S_{5} S_{2} S_{6} S_{3} S_{20} S_{19} S_{20} \\ S_{1} S_{1} S_{1} S_{11} S_{9} S_{12} S_{13} S_{15} S_{8} S_{16} S_{14} S_{12} \\ S_{5} S_{3} S_{5} S_{4} S_{6} S_{2} S_{2} S_{17} S_{7} \\ S_{1} S_{14} S_{9} S_{12} S_{15} S_{16} \\ S_{5} S_{2} S_{20} \\ S_{12} \\ \end{array} [/math] [math] \begin{array}{c} S_{13} \\ S_{3} S_{4} S_{19} \\ S_{1} S_{13} S_{11} S_{15} S_{8} S_{14} \\ S_{6} S_{3} S_{2} S_{5} S_{4} S_{3} S_{4} S_{7} S_{21} \\ S_{1} S_{1} S_{1} S_{10} S_{15} S_{13} S_{16} S_{12} S_{11} S_{14} S_{8} S_{13} \\ S_{4} S_{5} S_{5} S_{6} S_{3} S_{2} S_{3} S_{6} S_{4} S_{2} S_{19} S_{18} S_{19} \\ S_{1} S_{1} S_{1} S_{12} S_{8} S_{13} S_{15} S_{14} S_{10} S_{16} S_{11} S_{13} \\ S_{2} S_{3} S_{3} S_{5} S_{4} S_{6} S_{4} S_{21} S_{7} \\ S_{1} S_{15} S_{8} S_{13} S_{11} S_{14} \\ S_{4} S_{3} S_{19} \\ S_{13} \\ \end{array} [/math] [math] \begin{array}{c} S_{14} \\ S_{2} S_{4} S_{7} \\ S_{1} S_{16} S_{12} S_{14} S_{8} S_{13} \\ S_{3} S_{2} S_{4} S_{4} S_{5} S_{6} S_{2} S_{19} S_{18} \\ S_{1} S_{1} S_{1} S_{16} S_{15} S_{10} S_{13} S_{11} S_{14} S_{12} S_{8} S_{14} \\ S_{5} S_{2} S_{6} S_{4} S_{5} S_{3} S_{2} S_{4} S_{6} S_{3} S_{7} S_{21} S_{7} \\ S_{1} S_{1} S_{1} S_{8} S_{11} S_{15} S_{16} S_{13} S_{10} S_{12} S_{14} S_{14} \\ S_{6} S_{3} S_{4} S_{5} S_{4} S_{2} S_{2} S_{19} S_{18} \\ S_{1} S_{12} S_{8} S_{13} S_{16} S_{14} \\ S_{4} S_{2} S_{7} \\ S_{14} \\ \end{array} [/math] [math] \begin{array}{c} S_{15} \\ S_{3} S_{5} S_{17} \\ S_{1} S_{13} S_{9} S_{15} S_{11} S_{12} \\ S_{3} S_{2} S_{4} S_{3} S_{5} S_{6} S_{5} S_{20} S_{19} \\ S_{1} S_{1} S_{1} S_{15} S_{14} S_{13} S_{11} S_{16} S_{8} S_{12} S_{9} S_{15} \\ S_{6} S_{3} S_{5} S_{5} S_{4} S_{2} S_{4} S_{2} S_{6} S_{3} S_{7} S_{17} S_{17} \\ S_{1} S_{1} S_{1} S_{16} S_{15} S_{12} S_{14} S_{9} S_{8} S_{11} S_{15} S_{13} \\ S_{5} S_{2} S_{3} S_{5} S_{4} S_{3} S_{6} S_{19} S_{20} \\ S_{1} S_{12} S_{9} S_{11} S_{13} S_{15} \\ S_{5} S_{3} S_{17} \\ S_{15} \\ \end{array} [/math] [math] \begin{array}{c} S_{16} \\ S_{2} S_{6} S_{18} \\ S_{1} S_{12} S_{14} S_{11} S_{10} S_{16} \\ S_{3} S_{2} S_{4} S_{6} S_{5} S_{2} S_{6} S_{21} S_{20} \\ S_{1} S_{1} S_{1} S_{14} S_{13} S_{9} S_{11} S_{16} S_{15} S_{12} S_{10} S_{16} \\ S_{5} S_{2} S_{4} S_{6} S_{3} S_{3} S_{2} S_{6} S_{5} S_{4} S_{18} S_{17} S_{18} \\ S_{1} S_{1} S_{1} S_{16} S_{10} S_{13} S_{15} S_{11} S_{9} S_{16} S_{14} S_{12} \\ S_{3} S_{6} S_{6} S_{4} S_{5} S_{2} S_{2} S_{21} S_{20} \\ S_{1} S_{10} S_{11} S_{14} S_{16} S_{12} \\ S_{6} S_{2} S_{18} \\ S_{16} \\ \end{array} [/math] [math] \begin{array}{c} S_{17} \\ S_{15} S_{9} \\ S_{3} S_{5} S_{20} \\ S_{1} S_{12} S_{9} S_{11} \\ S_{2} S_{3} S_{6} S_{5} S_{17} \\ S_{1} S_{1} S_{16} S_{15} S_{15} \\ S_{2} S_{5} S_{3} S_{6} S_{17} \\ S_{1} S_{12} S_{11} S_{9} \\ S_{5} S_{3} S_{20} \\ S_{9} S_{15} \\ S_{17} \\ \end{array} [/math] [math] \begin{array}{c} S_{18} \\ S_{16} S_{10} \\ S_{6} S_{2} S_{21} \\ S_{1} S_{10} S_{14} S_{11} \\ S_{6} S_{3} S_{4} S_{2} S_{18} \\ S_{1} S_{1} S_{16} S_{16} S_{13} \\ S_{4} S_{3} S_{2} S_{6} S_{18} \\ S_{1} S_{11} S_{10} S_{14} \\ S_{2} S_{6} S_{21} \\ S_{10} S_{16} \\ S_{18} \\ \end{array} [/math] [math] \begin{array}{c} S_{19} \\ S_{8} S_{13} \\ S_{3} S_{4} S_{7} \\ S_{1} S_{14} S_{8} S_{15} \\ S_{5} S_{3} S_{2} S_{4} S_{19} \\ S_{1} S_{1} S_{13} S_{13} S_{12} \\ S_{4} S_{2} S_{5} S_{3} S_{19} \\ S_{1} S_{15} S_{14} S_{8} \\ S_{4} S_{3} S_{7} \\ S_{8} S_{13} \\ S_{19} \\ \end{array} [/math] [math] \begin{array}{c} S_{20} \\ S_{12} S_{9} \\ S_{2} S_{5} S_{17} \\ S_{1} S_{16} S_{9} S_{15} \\ S_{2} S_{3} S_{5} S_{6} S_{20} \\ S_{1} S_{1} S_{12} S_{12} S_{11} \\ S_{3} S_{5} S_{2} S_{6} S_{20} \\ S_{1} S_{9} S_{15} S_{16} \\ S_{5} S_{2} S_{17} \\ S_{9} S_{12} \\ S_{20} \\ \end{array} [/math] [math] \begin{array}{c} S_{21} \\ S_{11} S_{10} \\ S_{3} S_{6} S_{18} \\ S_{1} S_{13} S_{10} S_{16} \\ S_{4} S_{6} S_{2} S_{3} S_{21} \\ S_{1} S_{1} S_{14} S_{11} S_{11} \\ S_{4} S_{6} S_{3} S_{2} S_{21} \\ S_{1} S_{16} S_{13} S_{10} \\ S_{6} S_{3} S_{18} \\ S_{10} S_{11} \\ S_{21} \\ \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.
Cartan matrix
[math]\left( \begin{array}{ccccccccccccccccccccc} 32 & 16 & 16 & 16 & 16 & 16 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 & 4 & 4 \\ 16 & 16 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 8 & 8 & 4 & 8 & 4 & 4 & 2 & 2 & 4 & 4 & 2 & 4 \\ 16 & 8 & 16 & 8 & 8 & 8 & 4 & 4 & 4 & 4 & 4 & 8 & 4 & 8 & 8 & 4 & 4 & 2 & 2 & 4 & 2 \\ 16 & 8 & 8 & 16 & 8 & 8 & 4 & 8 & 0 & 8 & 4 & 8 & 4 & 4 & 4 & 2 & 4 & 4 & 0 & 0 & 2 \\ 16 & 8 & 8 & 8 & 16 & 8 & 0 & 4 & 8 & 4 & 4 & 4 & 8 & 8 & 4 & 0 & 2 & 2 & 4 & 4 & 0 \\ 16 & 8 & 8 & 8 & 8 & 16 & 8 & 0 & 4 & 4 & 8 & 4 & 4 & 4 & 8 & 4 & 0 & 0 & 2 & 2 & 4 \\ 8 & 4 & 4 & 4 & 0 & 8 & 8 & 0 & 0 & 2 & 4 & 2 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 0 & 4 \\ 8 & 4 & 4 & 8 & 4 & 0 & 0 & 8 & 0 & 4 & 0 & 4 & 2 & 2 & 0 & 0 & 4 & 4 & 0 & 0 & 0 \\ 8 & 4 & 4 & 0 & 8 & 4 & 0 & 0 & 8 & 0 & 2 & 0 & 4 & 4 & 2 & 0 & 0 & 0 & 4 & 4 & 0 \\ 8 & 8 & 4 & 8 & 4 & 4 & 2 & 4 & 0 & 8 & 4 & 4 & 4 & 2 & 2 & 1 & 2 & 4 & 0 & 0 & 2 \\ 8 & 8 & 4 & 4 & 4 & 8 & 4 & 0 & 2 & 4 & 8 & 2 & 4 & 2 & 4 & 2 & 0 & 0 & 2 & 1 & 4 \\ 8 & 4 & 8 & 8 & 4 & 4 & 2 & 4 & 0 & 4 & 2 & 8 & 2 & 4 & 4 & 2 & 4 & 2 & 0 & 0 & 1 \\ 8 & 8 & 4 & 4 & 8 & 4 & 0 & 2 & 4 & 4 & 4 & 2 & 8 & 4 & 2 & 0 & 1 & 2 & 4 & 2 & 0 \\ 8 & 4 & 8 & 4 & 8 & 4 & 0 & 2 & 4 & 2 & 2 & 4 & 4 & 8 & 4 & 0 & 2 & 1 & 2 & 4 & 0 \\ 8 & 4 & 8 & 4 & 4 & 8 & 4 & 0 & 2 & 2 & 4 & 4 & 2 & 4 & 8 & 4 & 0 & 0 & 1 & 2 & 2 \\ 4 & 2 & 4 & 2 & 0 & 4 & 4 & 0 & 0 & 1 & 2 & 2 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 0 & 2 \\ 4 & 2 & 4 & 4 & 2 & 0 & 0 & 4 & 0 & 2 & 0 & 4 & 1 & 2 & 0 & 0 & 4 & 2 & 0 & 0 & 0 \\ 4 & 4 & 2 & 4 & 2 & 0 & 0 & 4 & 0 & 4 & 0 & 2 & 2 & 1 & 0 & 0 & 2 & 4 & 0 & 0 & 0 \\ 4 & 4 & 2 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 2 & 0 & 4 & 2 & 1 & 0 & 0 & 0 & 4 & 2 & 0 \\ 4 & 2 & 4 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 1 & 0 & 2 & 4 & 2 & 0 & 0 & 0 & 2 & 4 & 0 \\ 4 & 4 & 2 & 2 & 0 & 4 & 4 & 0 & 0 & 2 & 4 & 1 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 4 \end{array}\right)[/math]
Decomposition matrix
[math]\left( \begin{array}{ccccccccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \end{array}\right)[/math]
