Difference between revisions of "M(32,51,15)"

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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(32,51,15), then <math>B</math> is in [[M(32,51,7)]], M(32,51,15), [[M(32,51,19)]], [[M(32,51,21)]] or [[M(32,51,33)]].
+
If <math>b</math> is in M(32,51,15), then <math>B</math> is in [[M(32,51,7)]], M(32,51,15), [[M(32,51,19)]] or [[M(32,51,34)]].
  
 
== Projective indecomposable modules ==
 
== Projective indecomposable modules ==
Line 46: Line 46:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{1} \\
 
S_{1} \\
S_{4} S_{6} S_{7} S_{8} S_{9} \\
+
S_{2} S_{3} S_{5} S_{4} S_{6} \\
S_{1} S_{1} S_{2} S_{3} S_{5} S_{21} S_{15} S_{19} S_{17} S_{20} S_{13} \\
+
S_{1} S_{1} S_{1} S_{1} S_{10} S_{9} S_{12} S_{11} S_{7} S_{8} S_{13} S_{14} S_{15} \\
S_{4} S_{6} S_{1} S_{7} S_{6} S_{4} S_{7} S_{8} S_{9} S_{18} S_{12} S_{10} S_{14} S_{16} S_{11} \\
+
S_{3} S_{3} S_{2} S_{2} S_{2} S_{3} S_{4} S_{6} S_{5} S_{6} S_{4} S_{6} S_{4} S_{5} S_{5} S_{21} S_{19} S_{16} S_{20} S_{17} S_{18} \\
S_{5} S_{3} S_{5} S_{1} S_{2} S_{3} S_{2} S_{13} S_{21} S_{15} S_{8} S_{9} S_{17} S_{20} S_{19} \\
+
S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{10} S_{9} S_{9} S_{10} S_{7} S_{12} S_{8} S_{8} S_{11} S_{12} S_{7} S_{11} S_{13} S_{15} S_{14} S_{15} S_{14} S_{13} \\
S_{7} S_{6} S_{4} S_{1} S_{1} S_{14} S_{10} S_{12} S_{16} S_{18} S_{11} \\
+
S_{2} S_{2} S_{2} S_{3} S_{3} S_{3} S_{5} S_{4} S_{6} S_{6} S_{5} S_{4} S_{4} S_{6} S_{5} S_{21} S_{17} S_{18} S_{20} S_{19} S_{16} \\
S_{2} S_{3} S_{5} S_{9} S_{8} \\
+
S_{1} S_{1} S_{1} S_{1} S_{12} S_{8} S_{10} S_{11} S_{9} S_{7} S_{14} S_{13} S_{15} \\
 +
S_{3} S_{2} S_{4} S_{6} S_{5} \\
 
S_{1} \\
 
S_{1} \\
 
   \end{array}
 
   \end{array}
Line 57: Line 58:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{2} \\
 
S_{2} \\
S_{4} S_{5} S_{1} S_{10} S_{14} \\
+
S_{1} S_{3} S_{7} S_{10} S_{11} \\
S_{2} S_{2} S_{6} S_{7} S_{3} S_{12} S_{18} S_{17} S_{9} S_{15} S_{8} \\
+
S_{2} S_{2} S_{2} S_{2} S_{9} S_{8} S_{4} S_{6} S_{5} S_{12} S_{17} S_{20} S_{21} \\
S_{5} S_{1} S_{4} S_{5} S_{1} S_{2} S_{4} S_{14} S_{13} S_{20} S_{21} S_{16} S_{19} S_{11} S_{10} \\
+
S_{1} S_{3} S_{3} S_{1} S_{1} S_{3} S_{7} S_{10} S_{10} S_{10} S_{11} S_{11} S_{7} S_{11} S_{7} S_{19} S_{14} S_{13} S_{18} S_{15} S_{16} \\
S_{6} S_{3} S_{6} S_{7} S_{7} S_{3} S_{2} S_{10} S_{14} S_{9} S_{8} S_{18} S_{17} S_{12} S_{15} \\
+
S_{2} S_{2} S_{2} S_{2} S_{2} S_{2} S_{6} S_{4} S_{6} S_{12} S_{12} S_{8} S_{9} S_{9} S_{5} S_{5} S_{4} S_{8} S_{20} S_{20} S_{21} S_{17} S_{17} S_{21} \\
S_{2} S_{2} S_{1} S_{5} S_{4} S_{11} S_{21} S_{13} S_{16} S_{20} S_{19} \\
+
S_{1} S_{1} S_{1} S_{3} S_{3} S_{3} S_{7} S_{10} S_{10} S_{7} S_{10} S_{11} S_{11} S_{7} S_{11} S_{19} S_{18} S_{16} S_{15} S_{13} S_{14} \\
S_{7} S_{6} S_{3} S_{10} S_{14} \\
+
S_{2} S_{2} S_{2} S_{2} S_{5} S_{12} S_{4} S_{6} S_{9} S_{8} S_{17} S_{20} S_{21} \\
 +
S_{3} S_{1} S_{10} S_{11} S_{7} \\
 
S_{2} \\
 
S_{2} \\
 
   \end{array}
 
   \end{array}
Line 68: Line 70:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{3} \\
 
S_{3} \\
S_{1} S_{2} S_{6} S_{11} S_{16} \\
+
S_{2} S_{1} S_{12} S_{9} S_{8} \\
S_{7} S_{3} S_{3} S_{4} S_{5} S_{14} S_{21} S_{13} S_{8} S_{9} S_{10} \\
+
S_{3} S_{3} S_{3} S_{3} S_{5} S_{7} S_{11} S_{10} S_{4} S_{6} S_{19} S_{18} S_{16} \\
S_{1} S_{3} S_{6} S_{2} S_{1} S_{2} S_{6} S_{11} S_{16} S_{19} S_{20} S_{15} S_{17} S_{18} S_{12} \\
+
S_{2} S_{1} S_{2} S_{2} S_{1} S_{1} S_{12} S_{12} S_{8} S_{8} S_{8} S_{9} S_{9} S_{12} S_{9} S_{14} S_{13} S_{20} S_{15} S_{17} S_{21} \\
S_{4} S_{7} S_{5} S_{7} S_{4} S_{5} S_{3} S_{16} S_{14} S_{10} S_{11} S_{13} S_{9} S_{8} S_{21} \\
+
S_{3} S_{3} S_{3} S_{3} S_{3} S_{3} S_{7} S_{4} S_{10} S_{5} S_{10} S_{11} S_{5} S_{6} S_{4} S_{6} S_{11} S_{7} S_{18} S_{18} S_{16} S_{19} S_{16} S_{19} \\
S_{1} S_{3} S_{6} S_{2} S_{3} S_{17} S_{19} S_{18} S_{12} S_{15} S_{20} \\
+
S_{1} S_{2} S_{1} S_{2} S_{2} S_{1} S_{9} S_{9} S_{8} S_{12} S_{9} S_{8} S_{8} S_{12} S_{12} S_{15} S_{14} S_{17} S_{13} S_{20} S_{21} \\
S_{7} S_{4} S_{5} S_{16} S_{11} \\
+
S_{3} S_{3} S_{3} S_{3} S_{6} S_{10} S_{5} S_{11} S_{4} S_{7} S_{16} S_{19} S_{18} \\
 +
S_{2} S_{1} S_{8} S_{9} S_{12} \\
 
S_{3} \\
 
S_{3} \\
 
   \end{array}
 
   \end{array}
Line 79: Line 82:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{4} \\
 
S_{4} \\
S_{6} S_{5} S_{3} S_{17} S_{15} \\
+
S_{1} S_{9} S_{7} S_{14} \\
S_{7} S_{1} S_{4} S_{2} S_{4} S_{16} S_{11} S_{18} S_{13} S_{21} S_{12} \\
+
S_{3} S_{2} S_{6} S_{4} S_{5} S_{4} S_{17} S_{19} \\
S_{4} S_{3} S_{3} S_{5} S_{6} S_{6} S_{5} S_{20} S_{10} S_{9} S_{19} S_{8} S_{14} S_{17} S_{15} \\
+
S_{1} S_{1} S_{1} S_{11} S_{7} S_{8} S_{12} S_{9} S_{10} S_{13} S_{14} \\
S_{7} S_{1} S_{2} S_{1} S_{2} S_{4} S_{7} S_{21} S_{11} S_{12} S_{13} S_{16} S_{17} S_{15} S_{18} \\
+
S_{2} S_{3} S_{3} S_{2} S_{6} S_{4} S_{5} S_{6} S_{4} S_{5} S_{18} S_{20} \\
S_{5} S_{3} S_{4} S_{6} S_{4} S_{14} S_{9} S_{8} S_{20} S_{10} S_{19} \\
+
S_{1} S_{1} S_{1} S_{8} S_{9} S_{10} S_{7} S_{12} S_{11} S_{13} S_{14} \\
S_{1} S_{7} S_{2} S_{17} S_{15} \\
+
S_{2} S_{3} S_{4} S_{4} S_{6} S_{5} S_{19} S_{17} \\
 +
S_{1} S_{7} S_{9} S_{14} \\
 
S_{4} \\
 
S_{4} \\
 
   \end{array}
 
   \end{array}
Line 95: Line 99:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{5} \\
 
S_{5} \\
S_{3} S_{7} S_{1} S_{18} S_{12} \\
+
S_{1} S_{11} S_{8} S_{15} \\
S_{6} S_{5} S_{5} S_{2} S_{4} S_{8} S_{19} S_{9} S_{16} S_{20} S_{11} \\
+
S_{2} S_{3} S_{6} S_{5} S_{5} S_{4} S_{16} S_{21} \\
S_{5} S_{1} S_{1} S_{3} S_{7} S_{7} S_{3} S_{13} S_{12} S_{21} S_{10} S_{18} S_{14} S_{17} S_{15} \\
+
S_{1} S_{1} S_{1} S_{10} S_{11} S_{8} S_{9} S_{12} S_{7} S_{14} S_{15} \\
S_{4} S_{2} S_{6} S_{6} S_{2} S_{4} S_{5} S_{8} S_{9} S_{11} S_{18} S_{12} S_{16} S_{19} S_{20} \\
+
S_{2} S_{3} S_{2} S_{3} S_{5} S_{5} S_{4} S_{6} S_{4} S_{6} S_{19} S_{17} \\
S_{1} S_{5} S_{7} S_{3} S_{5} S_{10} S_{21} S_{14} S_{15} S_{17} S_{13} \\
+
S_{1} S_{1} S_{1} S_{12} S_{8} S_{9} S_{11} S_{7} S_{10} S_{15} S_{14} \\
S_{2} S_{6} S_{4} S_{12} S_{18} \\
+
S_{3} S_{2} S_{5} S_{4} S_{5} S_{6} S_{21} S_{16} \\
 +
S_{1} S_{11} S_{8} S_{15} \\
 
S_{5} \\
 
S_{5} \\
 
   \end{array}
 
   \end{array}
Line 106: Line 111:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{6} \\
 
S_{6} \\
S_{5} S_{2} S_{7} S_{21} S_{13} \\
+
S_{1} S_{12} S_{10} S_{13} \\
S_{3} S_{1} S_{4} S_{6} S_{6} S_{12} S_{10} S_{18} S_{14} S_{20} S_{19} \\
+
S_{3} S_{2} S_{4} S_{6} S_{6} S_{5} S_{20} S_{18} \\
S_{2} S_{6} S_{5} S_{5} S_{7} S_{2} S_{7} S_{8} S_{15} S_{17} S_{11} S_{9} S_{16} S_{13} S_{21} \\
+
S_{1} S_{1} S_{1} S_{11} S_{7} S_{12} S_{9} S_{8} S_{10} S_{15} S_{13} \\
S_{3} S_{6} S_{4} S_{1} S_{3} S_{4} S_{1} S_{12} S_{13} S_{10} S_{20} S_{18} S_{14} S_{19} S_{21} \\
+
S_{2} S_{3} S_{2} S_{3} S_{4} S_{4} S_{6} S_{6} S_{5} S_{5} S_{16} S_{21} \\
S_{2} S_{7} S_{6} S_{6} S_{5} S_{8} S_{16} S_{17} S_{9} S_{15} S_{11} \\
+
S_{1} S_{1} S_{1} S_{7} S_{8} S_{11} S_{12} S_{10} S_{9} S_{15} S_{13} \\
S_{3} S_{1} S_{4} S_{21} S_{13} \\
+
S_{3} S_{2} S_{6} S_{6} S_{5} S_{4} S_{20} S_{18} \\
 +
S_{1} S_{10} S_{12} S_{13} \\
 
S_{6} \\
 
S_{6} \\
 
   \end{array}
 
   \end{array}
Line 117: Line 123:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{7} \\
 
S_{7} \\
S_{4} S_{2} S_{3} S_{19} S_{20} \\
+
S_{2} S_{9} S_{4} S_{17} \\
S_{5} S_{1} S_{7} S_{7} S_{6} S_{15} S_{16} S_{10} S_{14} S_{17} S_{11} \\
+
S_{3} S_{1} S_{7} S_{10} S_{11} S_{7} S_{19} S_{14} \\
S_{3} S_{2} S_{4} S_{7} S_{2} S_{4} S_{3} S_{9} S_{18} S_{8} S_{13} S_{21} S_{12} S_{19} S_{20} \\
+
S_{2} S_{2} S_{2} S_{12} S_{6} S_{5} S_{8} S_{9} S_{4} S_{17} S_{20} \\
S_{1} S_{6} S_{1} S_{5} S_{6} S_{7} S_{5} S_{10} S_{11} S_{14} S_{17} S_{16} S_{15} S_{19} S_{20} \\
+
S_{3} S_{3} S_{1} S_{1} S_{10} S_{7} S_{10} S_{11} S_{7} S_{11} S_{18} S_{13} \\
S_{4} S_{7} S_{3} S_{2} S_{7} S_{21} S_{9} S_{18} S_{13} S_{12} S_{8} \\
+
S_{2} S_{2} S_{2} S_{4} S_{8} S_{5} S_{12} S_{9} S_{6} S_{20} S_{17} \\
S_{5} S_{6} S_{1} S_{20} S_{19} \\
+
S_{3} S_{1} S_{11} S_{7} S_{10} S_{7} S_{14} S_{19} \\
 +
S_{2} S_{9} S_{4} S_{17} \\
 
S_{7} \\
 
S_{7} \\
 
   \end{array}
 
   \end{array}
Line 128: Line 135:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{8} \\
 
S_{8} \\
S_{1} S_{13} S_{20} S_{17} \\
+
S_{3} S_{11} S_{5} S_{16} \\
S_{4} S_{6} S_{7} S_{18} S_{11} S_{14} S_{9} \\
+
S_{1} S_{2} S_{9} S_{12} S_{8} S_{8} S_{15} S_{21} \\
S_{1} S_{5} S_{2} S_{3} S_{8} S_{21} S_{19} S_{15} \\
+
S_{3} S_{3} S_{3} S_{10} S_{7} S_{11} S_{4} S_{5} S_{6} S_{19} S_{16} \\
S_{1} S_{7} S_{4} S_{6} S_{10} S_{8} S_{12} S_{16} \\
+
S_{1} S_{1} S_{2} S_{2} S_{12} S_{9} S_{8} S_{8} S_{12} S_{9} S_{14} S_{17} \\
S_{2} S_{3} S_{5} S_{9} S_{17} S_{20} S_{13} \\
+
S_{3} S_{3} S_{3} S_{7} S_{10} S_{5} S_{4} S_{11} S_{6} S_{16} S_{19} \\
S_{1} S_{18} S_{11} S_{14} \\
+
S_{1} S_{2} S_{9} S_{8} S_{8} S_{12} S_{15} S_{21} \\
 +
S_{3} S_{5} S_{11} S_{16} \\
 
S_{8} \\
 
S_{8} \\
 
   \end{array}
 
   \end{array}
Line 145: Line 153:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{9} \\
 
S_{9} \\
S_{1} S_{19} S_{21} S_{15} \\
+
S_{3} S_{4} S_{7} S_{19} \\
S_{7} S_{6} S_{4} S_{12} S_{16} S_{8} S_{10} \\
+
S_{2} S_{1} S_{12} S_{9} S_{8} S_{9} S_{17} S_{14} \\
S_{5} S_{2} S_{1} S_{3} S_{9} S_{13} S_{20} S_{17} \\
+
S_{3} S_{3} S_{3} S_{5} S_{7} S_{4} S_{11} S_{10} S_{6} S_{19} S_{18} \\
S_{1} S_{7} S_{4} S_{6} S_{18} S_{11} S_{9} S_{14} \\
+
S_{1} S_{2} S_{2} S_{1} S_{9} S_{8} S_{8} S_{9} S_{12} S_{12} S_{13} S_{20} \\
S_{5} S_{3} S_{2} S_{8} S_{21} S_{15} S_{19} \\
+
S_{3} S_{3} S_{3} S_{6} S_{4} S_{11} S_{10} S_{5} S_{7} S_{19} S_{18} \\
S_{1} S_{16} S_{10} S_{12} \\
+
S_{1} S_{2} S_{9} S_{12} S_{8} S_{9} S_{17} S_{14} \\
 +
S_{3} S_{7} S_{4} S_{19} \\
 
S_{9} \\
 
S_{9} \\
 
   \end{array}
 
   \end{array}
Line 156: Line 165:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{10} \\
 
S_{10} \\
S_{2} S_{12} S_{15} S_{9} \\
+
S_{2} S_{12} S_{6} S_{20} \\
S_{1} S_{5} S_{4} S_{16} S_{21} S_{19} S_{14} \\
+
S_{1} S_{3} S_{11} S_{7} S_{10} S_{10} S_{18} S_{13} \\
S_{3} S_{2} S_{7} S_{6} S_{17} S_{10} S_{8} S_{18} \\
+
S_{2} S_{2} S_{2} S_{9} S_{5} S_{6} S_{4} S_{12} S_{8} S_{21} S_{20} \\
S_{1} S_{5} S_{2} S_{4} S_{13} S_{11} S_{20} S_{10} \\
+
S_{1} S_{1} S_{3} S_{3} S_{10} S_{11} S_{7} S_{7} S_{10} S_{11} S_{16} S_{15} \\
S_{6} S_{7} S_{3} S_{15} S_{9} S_{12} S_{14} \\
+
S_{2} S_{2} S_{2} S_{5} S_{4} S_{8} S_{6} S_{9} S_{12} S_{21} S_{20} \\
S_{2} S_{16} S_{21} S_{19} \\
+
S_{3} S_{1} S_{10} S_{7} S_{11} S_{10} S_{18} S_{13} \\
 +
S_{2} S_{12} S_{6} S_{20} \\
 
S_{10} \\
 
S_{10} \\
 
   \end{array}
 
   \end{array}
Line 167: Line 177:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{11} \\
 
S_{11} \\
S_{3} S_{13} S_{8} S_{14} \\
+
S_{2} S_{5} S_{8} S_{21} \\
S_{6} S_{2} S_{1} S_{16} S_{20} S_{17} S_{18} \\
+
S_{3} S_{1} S_{10} S_{11} S_{7} S_{11} S_{16} S_{15} \\
S_{7} S_{5} S_{3} S_{4} S_{11} S_{21} S_{10} S_{9} \\
+
S_{2} S_{2} S_{2} S_{4} S_{12} S_{6} S_{5} S_{8} S_{9} S_{21} S_{17} \\
S_{3} S_{1} S_{2} S_{6} S_{12} S_{15} S_{19} S_{11} \\
+
S_{3} S_{3} S_{1} S_{1} S_{7} S_{11} S_{10} S_{10} S_{7} S_{11} S_{19} S_{14} \\
S_{5} S_{7} S_{4} S_{13} S_{14} S_{8} S_{16} \\
+
S_{2} S_{2} S_{2} S_{9} S_{4} S_{12} S_{8} S_{6} S_{5} S_{17} S_{21} \\
S_{3} S_{20} S_{17} S_{18} \\
+
S_{1} S_{3} S_{10} S_{11} S_{11} S_{7} S_{15} S_{16} \\
 +
S_{2} S_{8} S_{5} S_{21} \\
 
S_{11} \\
 
S_{11} \\
 
   \end{array}
 
   \end{array}
Line 178: Line 189:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{12} \\
 
S_{12} \\
S_{5} S_{16} S_{9} S_{19} \\
+
S_{3} S_{10} S_{6} S_{18} \\
S_{7} S_{1} S_{3} S_{15} S_{18} S_{10} S_{21} \\
+
S_{2} S_{1} S_{12} S_{8} S_{9} S_{12} S_{20} S_{13} \\
S_{5} S_{2} S_{6} S_{4} S_{11} S_{8} S_{12} S_{20} \\
+
S_{3} S_{3} S_{3} S_{5} S_{10} S_{11} S_{4} S_{6} S_{7} S_{18} S_{16} \\
S_{3} S_{5} S_{1} S_{7} S_{14} S_{12} S_{13} S_{17} \\
+
S_{1} S_{1} S_{2} S_{2} S_{8} S_{9} S_{12} S_{12} S_{9} S_{8} S_{21} S_{15} \\
S_{4} S_{6} S_{2} S_{18} S_{9} S_{16} S_{19} \\
+
S_{3} S_{3} S_{3} S_{6} S_{10} S_{4} S_{7} S_{11} S_{5} S_{16} S_{18} \\
S_{5} S_{21} S_{15} S_{10} \\
+
S_{2} S_{1} S_{12} S_{12} S_{9} S_{8} S_{20} S_{13} \\
 +
S_{3} S_{10} S_{6} S_{18} \\
 
S_{12} \\
 
S_{12} \\
 
   \end{array}
 
   \end{array}
Line 195: Line 207:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{13} \\
 
S_{13} \\
S_{6} S_{20} S_{14} S_{18} \\
+
S_{6} S_{20} S_{18} \\
S_{5} S_{7} S_{2} S_{21} S_{17} S_{11} S_{8} \\
+
S_{1} S_{10} S_{12} S_{13} \\
S_{4} S_{6} S_{3} S_{1} S_{19} S_{12} S_{13} S_{10} \\
+
S_{2} S_{3} S_{4} S_{6} \\
S_{6} S_{2} S_{5} S_{7} S_{16} S_{9} S_{13} S_{15} \\
+
S_{1} S_{1} S_{7} S_{9} \\
S_{4} S_{3} S_{1} S_{18} S_{20} S_{14} S_{21} \\
+
S_{2} S_{3} S_{4} S_{6} \\
S_{6} S_{8} S_{11} S_{17} \\
+
S_{1} S_{12} S_{10} S_{13} \\
 +
S_{6} S_{18} S_{20} \\
 
S_{13} \\
 
S_{13} \\
 
   \end{array}
 
   \end{array}
Line 206: Line 219:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{14} \\
 
S_{14} \\
S_{2} S_{18} S_{17} S_{8} \\
+
S_{4} S_{19} S_{17} \\
S_{4} S_{5} S_{1} S_{13} S_{20} S_{11} S_{10} \\
+
S_{1} S_{9} S_{7} S_{14} \\
S_{7} S_{6} S_{3} S_{2} S_{14} S_{15} S_{12} S_{9} \\
+
S_{2} S_{3} S_{5} S_{4} \\
S_{5} S_{2} S_{1} S_{4} S_{14} S_{16} S_{19} S_{21} \\
+
S_{1} S_{1} S_{11} S_{8} \\
S_{3} S_{7} S_{6} S_{17} S_{10} S_{8} S_{18} \\
+
S_{2} S_{3} S_{5} S_{4} \\
S_{2} S_{20} S_{13} S_{11} \\
+
S_{1} S_{9} S_{7} S_{14} \\
 +
S_{4} S_{19} S_{17} \\
 
S_{14} \\
 
S_{14} \\
 
   \end{array}
 
   \end{array}
Line 217: Line 231:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{15} \\
 
S_{15} \\
S_{4} S_{21} S_{16} S_{12} \\
+
S_{5} S_{21} S_{16} \\
S_{6} S_{5} S_{3} S_{19} S_{9} S_{17} S_{10} \\
+
S_{1} S_{11} S_{8} S_{15} \\
S_{2} S_{4} S_{1} S_{7} S_{18} S_{11} S_{15} S_{13} \\
+
S_{2} S_{3} S_{5} S_{6} \\
S_{4} S_{5} S_{6} S_{3} S_{14} S_{20} S_{8} S_{15} \\
+
S_{1} S_{1} S_{10} S_{12} \\
S_{1} S_{7} S_{2} S_{16} S_{12} S_{17} S_{21} \\
+
S_{2} S_{3} S_{5} S_{6} \\
S_{4} S_{9} S_{10} S_{19} \\
+
S_{1} S_{8} S_{11} S_{15} \\
 +
S_{5} S_{16} S_{21} \\
 
S_{15} \\
 
S_{15} \\
 
   \end{array}
 
   \end{array}
Line 228: Line 243:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{16} \\
 
S_{16} \\
S_{3} S_{21} S_{10} S_{9} \\
+
S_{8} S_{15} S_{21} \\
S_{6} S_{1} S_{2} S_{19} S_{15} S_{12} S_{11} \\
+
S_{3} S_{5} S_{11} S_{16} \\
S_{5} S_{3} S_{7} S_{4} S_{16} S_{8} S_{14} S_{13} \\
+
S_{1} S_{2} S_{12} S_{8} \\
S_{6} S_{3} S_{1} S_{2} S_{16} S_{20} S_{18} S_{17} \\
+
S_{3} S_{3} S_{10} S_{6} \\
S_{7} S_{4} S_{5} S_{11} S_{21} S_{9} S_{10} \\
+
S_{2} S_{1} S_{8} S_{12} \\
S_{3} S_{19} S_{15} S_{12} \\
+
S_{3} S_{11} S_{5} S_{16} \\
 +
S_{8} S_{15} S_{21} \\
 
S_{16} \\
 
S_{16} \\
 
   \end{array}
 
   \end{array}
Line 245: Line 261:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{17} \\
 
S_{17} \\
S_{4} S_{13} S_{18} S_{11} \\
+
S_{7} S_{19} S_{14} \\
S_{3} S_{6} S_{5} S_{14} S_{20} S_{15} S_{8} \\
+
S_{2} S_{4} S_{9} S_{17} \\
S_{2} S_{1} S_{4} S_{7} S_{21} S_{12} S_{16} S_{17} \\
+
S_{3} S_{1} S_{11} S_{7} \\
S_{4} S_{6} S_{5} S_{3} S_{19} S_{17} S_{9} S_{10} \\
+
S_{2} S_{2} S_{5} S_{8} \\
S_{2} S_{7} S_{1} S_{18} S_{15} S_{13} S_{11} \\
+
S_{1} S_{3} S_{11} S_{7} \\
S_{4} S_{20} S_{14} S_{8} \\
+
S_{2} S_{4} S_{9} S_{17} \\
 +
S_{7} S_{14} S_{19} \\
 
S_{17} \\
 
S_{17} \\
 
   \end{array}
 
   \end{array}
Line 256: Line 273:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{18} \\
 
S_{18} \\
S_{5} S_{11} S_{8} S_{20} \\
+
S_{12} S_{13} S_{20} \\
S_{7} S_{3} S_{1} S_{12} S_{17} S_{14} S_{13} \\
+
S_{3} S_{10} S_{6} S_{18} \\
S_{2} S_{4} S_{5} S_{6} S_{19} S_{9} S_{16} S_{18} \\
+
S_{1} S_{2} S_{12} S_{9} \\
S_{1} S_{5} S_{7} S_{3} S_{21} S_{18} S_{10} S_{15} \\
+
S_{3} S_{3} S_{7} S_{4} \\
S_{6} S_{4} S_{2} S_{12} S_{8} S_{11} S_{20} \\
+
S_{1} S_{2} S_{12} S_{9} \\
S_{5} S_{17} S_{14} S_{13} \\
+
S_{3} S_{6} S_{10} S_{18} \\
 +
S_{12} S_{13} S_{20} \\
 
S_{18} \\
 
S_{18} \\
 
   \end{array}
 
   \end{array}
Line 267: Line 285:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{19} \\
 
S_{19} \\
S_{7} S_{10} S_{16} S_{15} \\
+
S_{9} S_{14} S_{17} \\
S_{2} S_{4} S_{3} S_{12} S_{21} S_{9} S_{20} \\
+
S_{3} S_{7} S_{4} S_{19} \\
S_{1} S_{6} S_{7} S_{5} S_{17} S_{11} S_{14} S_{19} \\
+
S_{1} S_{2} S_{9} S_{8} \\
S_{2} S_{4} S_{7} S_{3} S_{13} S_{8} S_{19} S_{18} \\
+
S_{3} S_{3} S_{5} S_{11} \\
S_{6} S_{1} S_{5} S_{16} S_{20} S_{10} S_{15} \\
+
S_{1} S_{2} S_{9} S_{8} \\
S_{7} S_{21} S_{9} S_{12} \\
+
S_{3} S_{7} S_{4} S_{19} \\
 +
S_{9} S_{14} S_{17} \\
 
S_{19} \\
 
S_{19} \\
 
   \end{array}
 
   \end{array}
Line 278: Line 297:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{20} \\
 
S_{20} \\
S_{7} S_{11} S_{17} S_{14} \\
+
S_{10} S_{18} S_{13} \\
S_{2} S_{4} S_{3} S_{19} S_{8} S_{13} S_{18} \\
+
S_{2} S_{12} S_{6} S_{20} \\
S_{6} S_{5} S_{1} S_{7} S_{16} S_{10} S_{15} S_{20} \\
+
S_{1} S_{3} S_{10} S_{7} \\
S_{7} S_{4} S_{2} S_{3} S_{9} S_{20} S_{12} S_{21} \\
+
S_{2} S_{2} S_{9} S_{4} \\
S_{1} S_{6} S_{5} S_{17} S_{14} S_{19} S_{11} \\
+
S_{3} S_{1} S_{7} S_{10} \\
S_{7} S_{13} S_{18} S_{8} \\
+
S_{2} S_{6} S_{12} S_{20} \\
 +
S_{10} S_{18} S_{13} \\
 
S_{20} \\
 
S_{20} \\
 
   \end{array}
 
   \end{array}
Line 289: Line 309:
 
\begin{array}{c}
 
\begin{array}{c}
 
S_{21} \\
 
S_{21} \\
S_{6} S_{10} S_{12} S_{19} \\
+
S_{11} S_{16} S_{15} \\
S_{2} S_{7} S_{5} S_{15} S_{9} S_{13} S_{16} \\
+
S_{2} S_{5} S_{8} S_{21} \\
S_{1} S_{3} S_{4} S_{6} S_{14} S_{18} S_{21} S_{20} \\
+
S_{1} S_{3} S_{11} S_{10} \\
S_{7} S_{2} S_{5} S_{6} S_{11} S_{8} S_{17} S_{21} \\
+
S_{2} S_{2} S_{12} S_{6} \\
S_{3} S_{1} S_{4} S_{19} S_{10} S_{12} S_{13} \\
+
S_{3} S_{1} S_{11} S_{10} \\
S_{6} S_{9} S_{16} S_{15} \\
+
S_{2} S_{8} S_{5} S_{21} \\
 +
S_{11} S_{16} S_{15} \\
 
S_{21} \\
 
S_{21} \\
 
   \end{array}
 
   \end{array}
Line 306: Line 327:
 
== Cartan matrix ==
 
== Cartan matrix ==
 
<math>\left( \begin{array}{ccccccccccccccccccccc}
 
<math>\left( \begin{array}{ccccccccccccccccccccc}
8 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
+
16 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\
4 & 8 & 4 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 \\
+
8 & 16 & 8 & 4 & 4 & 4 & 4 & 8 & 8 & 4 & 4 & 8 & 2 & 2 & 2 & 2 & 2 & 4 & 4 & 2 & 4 \\
4 & 4 & 8 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 \\
+
8 & 8 & 16 & 4 & 4 & 4 & 8 & 4 & 4 & 8 & 8 & 4 & 2 & 2 & 2 & 4 & 4 & 2 & 2 & 4 & 2 \\
4 & 4 & 4 & 8 & 4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 4 & 2 \\
+
8 & 4 & 4 & 8 & 4 & 4 & 4 & 2 & 2 & 2 & 2 & 4 & 2 & 4 & 0 & 0 & 2 & 1 & 2 & 1 & 0 \\
4 & 4 & 4 & 4 & 8 & 4 & 4 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\
+
8 & 4 & 4 & 4 & 8 & 4 & 2 & 4 & 2 & 4 & 2 & 2 & 0 & 2 & 4 & 2 & 1 & 0 & 1 & 0 & 2 \\
4 & 4 & 4 & 4 & 4 & 8 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 4 \\
+
8 & 4 & 4 & 4 & 4 & 8 & 2 & 2 & 4 & 2 & 4 & 2 & 4 & 0 & 2 & 1 & 0 & 2 & 0 & 2 & 1 \\
4 & 4 & 4 & 4 & 4 & 4 & 8 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 \\
+
4 & 4 & 8 & 4 & 2 & 2 & 8 & 2 & 2 & 4 & 4 & 4 & 1 & 2 & 0 & 0 & 4 & 1 & 2 & 2 & 0 \\
4 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 2 \\
+
4 & 8 & 4 & 2 & 4 & 2 & 2 & 8 & 4 & 4 & 2 & 4 & 0 & 1 & 2 & 2 & 1 & 0 & 2 & 0 & 4 \\
4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 1 \\
+
4 & 8 & 4 & 2 & 2 & 4 & 2 & 4 & 8 & 2 & 4 & 4 & 2 & 0 & 1 & 1 & 0 & 4 & 0 & 2 & 2 \\
2 & 2 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 4 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 2 & 1 \\
+
4 & 4 & 8 & 2 & 4 & 2 & 4 & 4 & 2 & 8 & 4 & 2 & 0 & 1 & 2 & 4 & 2 & 0 & 1 & 0 & 2 \\
2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 4 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 2 \\
+
4 & 4 & 8 & 2 & 2 & 4 & 4 & 2 & 4 & 4 & 8 & 2 & 2 & 0 & 1 & 2 & 0 & 2 & 0 & 4 & 1 \\
2 & 2 & 2 & 2 & 2 & 2 & 4 & 1 & 2 & 2 & 1 & 4 & 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 1 \\
+
4 & 8 & 4 & 4 & 2 & 2 & 4 & 4 & 4 & 2 & 2 & 8 & 1 & 2 & 0 & 0 & 2 & 2 & 4 & 1 & 0 \\
2 & 2 & 4 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 4 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 1 \\
+
4 & 2 & 2 & 2 & 0 & 4 & 1 & 0 & 2 & 0 & 2 & 1 & 4 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 \\
2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 1 & 4 & 2 & 1 & 1 & 2 & 2 & 2 & 2 \\
+
4 & 2 & 2 & 4 & 2 & 0 & 2 & 1 & 0 & 1 & 0 & 2 & 0 & 4 & 0 & 0 & 2 & 0 & 2 & 0 & 0 \\
2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 & 2 & 1 & 2 \\
+
4 & 2 & 2 & 0 & 4 & 2 & 0 & 2 & 1 & 2 & 1 & 0 & 0 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 2 \\
2 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 4 & 2 & 1 & 1 & 2 & 2 \\
+
2 & 2 & 4 & 0 & 2 & 1 & 0 & 2 & 1 & 4 & 2 & 0 & 0 & 0 & 2 & 4 & 0 & 0 & 0 & 0 & 2 \\
2 & 4 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 & 1 \\
+
2 & 2 & 4 & 2 & 1 & 0 & 4 & 1 & 0 & 2 & 0 & 2 & 0 & 2 & 0 & 0 & 4 & 0 & 2 & 0 & 0 \\
2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 4 & 2 & 1 & 2 \\
+
2 & 4 & 2 & 1 & 0 & 2 & 1 & 0 & 4 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 2 & 0 \\
2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 2 & 4 & 1 & 2 \\
+
2 & 4 & 2 & 2 & 1 & 0 & 2 & 2 & 0 & 1 & 0 & 4 & 0 & 2 & 0 & 0 & 2 & 0 & 4 & 0 & 0 \\
2 & 2 & 2 & 4 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 4 & 1 \\
+
2 & 2 & 4 & 1 & 0 & 2 & 2 & 0 & 2 & 0 & 4 & 1 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 4 & 0 \\
2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 4
+
2 & 4 & 2 & 0 & 2 & 1 & 0 & 4 & 2 & 2 & 1 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 4  
 
\end{array}\right)</math>
 
\end{array}\right)</math>
  
Line 333: Line 354:
 
<math>\left( \begin{array}{ccccccccccccccccccccc}
 
<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 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 
 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 
 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 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 \\
0 & 0 & 0 & 0 & 1 & 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 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
+
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 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 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
+
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
+
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
+
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
+
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
+
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
+
0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
+
0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
+
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
+
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
+
1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 \\
+
1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\
+
1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 &
+
1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
 +
1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\
 +
1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0
 
\end{array}\right)</math>
 
\end{array}\right)</math>
  
 
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]
 
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]

Revision as of 15:09, 8 December 2019

M(32,51,15) - [math]B_0(k(SL_2(8) \times A_4))[/math]
[[File: |250px]]
Representative: [math]B_0(k(SL_2(8) \times A_4))[/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_4))[/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(32,51,16)
[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:


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(32,51,15), then [math]B[/math] is in M(32,51,7), M(32,51,15), M(32,51,19) or M(32,51,34).

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}{cccc} \begin{array}{c} S_{1} \\ S_{2} S_{3} S_{5} S_{4} S_{6} \\ S_{1} S_{1} S_{1} S_{1} S_{10} S_{9} S_{12} S_{11} S_{7} S_{8} S_{13} S_{14} S_{15} \\ S_{3} S_{3} S_{2} S_{2} S_{2} S_{3} S_{4} S_{6} S_{5} S_{6} S_{4} S_{6} S_{4} S_{5} S_{5} S_{21} S_{19} S_{16} S_{20} S_{17} S_{18} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{10} S_{9} S_{9} S_{10} S_{7} S_{12} S_{8} S_{8} S_{11} S_{12} S_{7} S_{11} S_{13} S_{15} S_{14} S_{15} S_{14} S_{13} \\ S_{2} S_{2} S_{2} S_{3} S_{3} S_{3} S_{5} S_{4} S_{6} S_{6} S_{5} S_{4} S_{4} S_{6} S_{5} S_{21} S_{17} S_{18} S_{20} S_{19} S_{16} \\ S_{1} S_{1} S_{1} S_{1} S_{12} S_{8} S_{10} S_{11} S_{9} S_{7} S_{14} S_{13} S_{15} \\ S_{3} S_{2} S_{4} S_{6} S_{5} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{1} S_{3} S_{7} S_{10} S_{11} \\ S_{2} S_{2} S_{2} S_{2} S_{9} S_{8} S_{4} S_{6} S_{5} S_{12} S_{17} S_{20} S_{21} \\ S_{1} S_{3} S_{3} S_{1} S_{1} S_{3} S_{7} S_{10} S_{10} S_{10} S_{11} S_{11} S_{7} S_{11} S_{7} S_{19} S_{14} S_{13} S_{18} S_{15} S_{16} \\ S_{2} S_{2} S_{2} S_{2} S_{2} S_{2} S_{6} S_{4} S_{6} S_{12} S_{12} S_{8} S_{9} S_{9} S_{5} S_{5} S_{4} S_{8} S_{20} S_{20} S_{21} S_{17} S_{17} S_{21} \\ S_{1} S_{1} S_{1} S_{3} S_{3} S_{3} S_{7} S_{10} S_{10} S_{7} S_{10} S_{11} S_{11} S_{7} S_{11} S_{19} S_{18} S_{16} S_{15} S_{13} S_{14} \\ S_{2} S_{2} S_{2} S_{2} S_{5} S_{12} S_{4} S_{6} S_{9} S_{8} S_{17} S_{20} S_{21} \\ S_{3} S_{1} S_{10} S_{11} S_{7} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{2} S_{1} S_{12} S_{9} S_{8} \\ S_{3} S_{3} S_{3} S_{3} S_{5} S_{7} S_{11} S_{10} S_{4} S_{6} S_{19} S_{18} S_{16} \\ S_{2} S_{1} S_{2} S_{2} S_{1} S_{1} S_{12} S_{12} S_{8} S_{8} S_{8} S_{9} S_{9} S_{12} S_{9} S_{14} S_{13} S_{20} S_{15} S_{17} S_{21} \\ S_{3} S_{3} S_{3} S_{3} S_{3} S_{3} S_{7} S_{4} S_{10} S_{5} S_{10} S_{11} S_{5} S_{6} S_{4} S_{6} S_{11} S_{7} S_{18} S_{18} S_{16} S_{19} S_{16} S_{19} \\ S_{1} S_{2} S_{1} S_{2} S_{2} S_{1} S_{9} S_{9} S_{8} S_{12} S_{9} S_{8} S_{8} S_{12} S_{12} S_{15} S_{14} S_{17} S_{13} S_{20} S_{21} \\ S_{3} S_{3} S_{3} S_{3} S_{6} S_{10} S_{5} S_{11} S_{4} S_{7} S_{16} S_{19} S_{18} \\ S_{2} S_{1} S_{8} S_{9} S_{12} \\ S_{3} \\ \end{array} & \begin{array}{c} S_{4} \\ S_{1} S_{9} S_{7} S_{14} \\ S_{3} S_{2} S_{6} S_{4} S_{5} S_{4} S_{17} S_{19} \\ S_{1} S_{1} S_{1} S_{11} S_{7} S_{8} S_{12} S_{9} S_{10} S_{13} S_{14} \\ S_{2} S_{3} S_{3} S_{2} S_{6} S_{4} S_{5} S_{6} S_{4} S_{5} S_{18} S_{20} \\ S_{1} S_{1} S_{1} S_{8} S_{9} S_{10} S_{7} S_{12} S_{11} S_{13} S_{14} \\ S_{2} S_{3} S_{4} S_{4} S_{6} S_{5} S_{19} S_{17} \\ S_{1} S_{7} S_{9} S_{14} \\ S_{4} \\ \end{array} \end{array}[/math]


 

[math] \begin{array}{cccc} \begin{array}{c} S_{5} \\ S_{1} S_{11} S_{8} S_{15} \\ S_{2} S_{3} S_{6} S_{5} S_{5} S_{4} S_{16} S_{21} \\ S_{1} S_{1} S_{1} S_{10} S_{11} S_{8} S_{9} S_{12} S_{7} S_{14} S_{15} \\ S_{2} S_{3} S_{2} S_{3} S_{5} S_{5} S_{4} S_{6} S_{4} S_{6} S_{19} S_{17} \\ S_{1} S_{1} S_{1} S_{12} S_{8} S_{9} S_{11} S_{7} S_{10} S_{15} S_{14} \\ S_{3} S_{2} S_{5} S_{4} S_{5} S_{6} S_{21} S_{16} \\ S_{1} S_{11} S_{8} S_{15} \\ S_{5} \\ \end{array} & \begin{array}{c} S_{6} \\ S_{1} S_{12} S_{10} S_{13} \\ S_{3} S_{2} S_{4} S_{6} S_{6} S_{5} S_{20} S_{18} \\ S_{1} S_{1} S_{1} S_{11} S_{7} S_{12} S_{9} S_{8} S_{10} S_{15} S_{13} \\ S_{2} S_{3} S_{2} S_{3} S_{4} S_{4} S_{6} S_{6} S_{5} S_{5} S_{16} S_{21} \\ S_{1} S_{1} S_{1} S_{7} S_{8} S_{11} S_{12} S_{10} S_{9} S_{15} S_{13} \\ S_{3} S_{2} S_{6} S_{6} S_{5} S_{4} S_{20} S_{18} \\ S_{1} S_{10} S_{12} S_{13} \\ S_{6} \\ \end{array} & \begin{array}{c} S_{7} \\ S_{2} S_{9} S_{4} S_{17} \\ S_{3} S_{1} S_{7} S_{10} S_{11} S_{7} S_{19} S_{14} \\ S_{2} S_{2} S_{2} S_{12} S_{6} S_{5} S_{8} S_{9} S_{4} S_{17} S_{20} \\ S_{3} S_{3} S_{1} S_{1} S_{10} S_{7} S_{10} S_{11} S_{7} S_{11} S_{18} S_{13} \\ S_{2} S_{2} S_{2} S_{4} S_{8} S_{5} S_{12} S_{9} S_{6} S_{20} S_{17} \\ S_{3} S_{1} S_{11} S_{7} S_{10} S_{7} S_{14} S_{19} \\ S_{2} S_{9} S_{4} S_{17} \\ S_{7} \\ \end{array} & \begin{array}{c} S_{8} \\ S_{3} S_{11} S_{5} S_{16} \\ S_{1} S_{2} S_{9} S_{12} S_{8} S_{8} S_{15} S_{21} \\ S_{3} S_{3} S_{3} S_{10} S_{7} S_{11} S_{4} S_{5} S_{6} S_{19} S_{16} \\ S_{1} S_{1} S_{2} S_{2} S_{12} S_{9} S_{8} S_{8} S_{12} S_{9} S_{14} S_{17} \\ S_{3} S_{3} S_{3} S_{7} S_{10} S_{5} S_{4} S_{11} S_{6} S_{16} S_{19} \\ S_{1} S_{2} S_{9} S_{8} S_{8} S_{12} S_{15} S_{21} \\ S_{3} S_{5} S_{11} S_{16} \\ S_{8} \\ \end{array} \end{array} [/math]


 

[math] \begin{array}{cccc} \begin{array}{c} S_{9} \\ S_{3} S_{4} S_{7} S_{19} \\ S_{2} S_{1} S_{12} S_{9} S_{8} S_{9} S_{17} S_{14} \\ S_{3} S_{3} S_{3} S_{5} S_{7} S_{4} S_{11} S_{10} S_{6} S_{19} S_{18} \\ S_{1} S_{2} S_{2} S_{1} S_{9} S_{8} S_{8} S_{9} S_{12} S_{12} S_{13} S_{20} \\ S_{3} S_{3} S_{3} S_{6} S_{4} S_{11} S_{10} S_{5} S_{7} S_{19} S_{18} \\ S_{1} S_{2} S_{9} S_{12} S_{8} S_{9} S_{17} S_{14} \\ S_{3} S_{7} S_{4} S_{19} \\ S_{9} \\ \end{array} & \begin{array}{c} S_{10} \\ S_{2} S_{12} S_{6} S_{20} \\ S_{1} S_{3} S_{11} S_{7} S_{10} S_{10} S_{18} S_{13} \\ S_{2} S_{2} S_{2} S_{9} S_{5} S_{6} S_{4} S_{12} S_{8} S_{21} S_{20} \\ S_{1} S_{1} S_{3} S_{3} S_{10} S_{11} S_{7} S_{7} S_{10} S_{11} S_{16} S_{15} \\ S_{2} S_{2} S_{2} S_{5} S_{4} S_{8} S_{6} S_{9} S_{12} S_{21} S_{20} \\ S_{3} S_{1} S_{10} S_{7} S_{11} S_{10} S_{18} S_{13} \\ S_{2} S_{12} S_{6} S_{20} \\ S_{10} \\ \end{array} & \begin{array}{c} S_{11} \\ S_{2} S_{5} S_{8} S_{21} \\ S_{3} S_{1} S_{10} S_{11} S_{7} S_{11} S_{16} S_{15} \\ S_{2} S_{2} S_{2} S_{4} S_{12} S_{6} S_{5} S_{8} S_{9} S_{21} S_{17} \\ S_{3} S_{3} S_{1} S_{1} S_{7} S_{11} S_{10} S_{10} S_{7} S_{11} S_{19} S_{14} \\ S_{2} S_{2} S_{2} S_{9} S_{4} S_{12} S_{8} S_{6} S_{5} S_{17} S_{21} \\ S_{1} S_{3} S_{10} S_{11} S_{11} S_{7} S_{15} S_{16} \\ S_{2} S_{8} S_{5} S_{21} \\ S_{11} \\ \end{array} & \begin{array}{c} S_{12} \\ S_{3} S_{10} S_{6} S_{18} \\ S_{2} S_{1} S_{12} S_{8} S_{9} S_{12} S_{20} S_{13} \\ S_{3} S_{3} S_{3} S_{5} S_{10} S_{11} S_{4} S_{6} S_{7} S_{18} S_{16} \\ S_{1} S_{1} S_{2} S_{2} S_{8} S_{9} S_{12} S_{12} S_{9} S_{8} S_{21} S_{15} \\ S_{3} S_{3} S_{3} S_{6} S_{10} S_{4} S_{7} S_{11} S_{5} S_{16} S_{18} \\ S_{2} S_{1} S_{12} S_{12} S_{9} S_{8} S_{20} S_{13} \\ S_{3} S_{10} S_{6} S_{18} \\ S_{12} \\ \end{array} \end{array} [/math]


 

[math] \begin{array}{cccc} \begin{array}{c} S_{13} \\ S_{6} S_{20} S_{18} \\ S_{1} S_{10} S_{12} S_{13} \\ S_{2} S_{3} S_{4} S_{6} \\ S_{1} S_{1} S_{7} S_{9} \\ S_{2} S_{3} S_{4} S_{6} \\ S_{1} S_{12} S_{10} S_{13} \\ S_{6} S_{18} S_{20} \\ S_{13} \\ \end{array} & \begin{array}{c} S_{14} \\ S_{4} S_{19} S_{17} \\ S_{1} S_{9} S_{7} S_{14} \\ S_{2} S_{3} S_{5} S_{4} \\ S_{1} S_{1} S_{11} S_{8} \\ S_{2} S_{3} S_{5} S_{4} \\ S_{1} S_{9} S_{7} S_{14} \\ S_{4} S_{19} S_{17} \\ S_{14} \\ \end{array} & \begin{array}{c} S_{15} \\ S_{5} S_{21} S_{16} \\ S_{1} S_{11} S_{8} S_{15} \\ S_{2} S_{3} S_{5} S_{6} \\ S_{1} S_{1} S_{10} S_{12} \\ S_{2} S_{3} S_{5} S_{6} \\ S_{1} S_{8} S_{11} S_{15} \\ S_{5} S_{16} S_{21} \\ S_{15} \\ \end{array} & \begin{array}{c} S_{16} \\ S_{8} S_{15} S_{21} \\ S_{3} S_{5} S_{11} S_{16} \\ S_{1} S_{2} S_{12} S_{8} \\ S_{3} S_{3} S_{10} S_{6} \\ S_{2} S_{1} S_{8} S_{12} \\ S_{3} S_{11} S_{5} S_{16} \\ S_{8} S_{15} S_{21} \\ S_{16} \\ \end{array} \end{array} [/math]


 

[math] \begin{array}{ccccc} \begin{array}{c} S_{17} \\ S_{7} S_{19} S_{14} \\ S_{2} S_{4} S_{9} S_{17} \\ S_{3} S_{1} S_{11} S_{7} \\ S_{2} S_{2} S_{5} S_{8} \\ S_{1} S_{3} S_{11} S_{7} \\ S_{2} S_{4} S_{9} S_{17} \\ S_{7} S_{14} S_{19} \\ S_{17} \\ \end{array} & \begin{array}{c} S_{18} \\ S_{12} S_{13} S_{20} \\ S_{3} S_{10} S_{6} S_{18} \\ S_{1} S_{2} S_{12} S_{9} \\ S_{3} S_{3} S_{7} S_{4} \\ S_{1} S_{2} S_{12} S_{9} \\ S_{3} S_{6} S_{10} S_{18} \\ S_{12} S_{13} S_{20} \\ S_{18} \\ \end{array} & \begin{array}{c} S_{19} \\ S_{9} S_{14} S_{17} \\ S_{3} S_{7} S_{4} S_{19} \\ S_{1} S_{2} S_{9} S_{8} \\ S_{3} S_{3} S_{5} S_{11} \\ S_{1} S_{2} S_{9} S_{8} \\ S_{3} S_{7} S_{4} S_{19} \\ S_{9} S_{14} S_{17} \\ S_{19} \\ \end{array} & \begin{array}{c} S_{20} \\ S_{10} S_{18} S_{13} \\ S_{2} S_{12} S_{6} S_{20} \\ S_{1} S_{3} S_{10} S_{7} \\ S_{2} S_{2} S_{9} S_{4} \\ S_{3} S_{1} S_{7} S_{10} \\ S_{2} S_{6} S_{12} S_{20} \\ S_{10} S_{18} S_{13} \\ S_{20} \\ \end{array} & \begin{array}{c} S_{21} \\ S_{11} S_{16} S_{15} \\ S_{2} S_{5} S_{8} S_{21} \\ S_{1} S_{3} S_{11} S_{10} \\ S_{2} S_{2} S_{12} S_{6} \\ S_{3} S_{1} S_{11} S_{10} \\ S_{2} S_{8} S_{5} S_{21} \\ S_{11} S_{16} S_{15} \\ S_{21} \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Cartan matrix

[math]\left( \begin{array}{ccccccccccccccccccccc} 16 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\ 8 & 16 & 8 & 4 & 4 & 4 & 4 & 8 & 8 & 4 & 4 & 8 & 2 & 2 & 2 & 2 & 2 & 4 & 4 & 2 & 4 \\ 8 & 8 & 16 & 4 & 4 & 4 & 8 & 4 & 4 & 8 & 8 & 4 & 2 & 2 & 2 & 4 & 4 & 2 & 2 & 4 & 2 \\ 8 & 4 & 4 & 8 & 4 & 4 & 4 & 2 & 2 & 2 & 2 & 4 & 2 & 4 & 0 & 0 & 2 & 1 & 2 & 1 & 0 \\ 8 & 4 & 4 & 4 & 8 & 4 & 2 & 4 & 2 & 4 & 2 & 2 & 0 & 2 & 4 & 2 & 1 & 0 & 1 & 0 & 2 \\ 8 & 4 & 4 & 4 & 4 & 8 & 2 & 2 & 4 & 2 & 4 & 2 & 4 & 0 & 2 & 1 & 0 & 2 & 0 & 2 & 1 \\ 4 & 4 & 8 & 4 & 2 & 2 & 8 & 2 & 2 & 4 & 4 & 4 & 1 & 2 & 0 & 0 & 4 & 1 & 2 & 2 & 0 \\ 4 & 8 & 4 & 2 & 4 & 2 & 2 & 8 & 4 & 4 & 2 & 4 & 0 & 1 & 2 & 2 & 1 & 0 & 2 & 0 & 4 \\ 4 & 8 & 4 & 2 & 2 & 4 & 2 & 4 & 8 & 2 & 4 & 4 & 2 & 0 & 1 & 1 & 0 & 4 & 0 & 2 & 2 \\ 4 & 4 & 8 & 2 & 4 & 2 & 4 & 4 & 2 & 8 & 4 & 2 & 0 & 1 & 2 & 4 & 2 & 0 & 1 & 0 & 2 \\ 4 & 4 & 8 & 2 & 2 & 4 & 4 & 2 & 4 & 4 & 8 & 2 & 2 & 0 & 1 & 2 & 0 & 2 & 0 & 4 & 1 \\ 4 & 8 & 4 & 4 & 2 & 2 & 4 & 4 & 4 & 2 & 2 & 8 & 1 & 2 & 0 & 0 & 2 & 2 & 4 & 1 & 0 \\ 4 & 2 & 2 & 2 & 0 & 4 & 1 & 0 & 2 & 0 & 2 & 1 & 4 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 \\ 4 & 2 & 2 & 4 & 2 & 0 & 2 & 1 & 0 & 1 & 0 & 2 & 0 & 4 & 0 & 0 & 2 & 0 & 2 & 0 & 0 \\ 4 & 2 & 2 & 0 & 4 & 2 & 0 & 2 & 1 & 2 & 1 & 0 & 0 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 2 \\ 2 & 2 & 4 & 0 & 2 & 1 & 0 & 2 & 1 & 4 & 2 & 0 & 0 & 0 & 2 & 4 & 0 & 0 & 0 & 0 & 2 \\ 2 & 2 & 4 & 2 & 1 & 0 & 4 & 1 & 0 & 2 & 0 & 2 & 0 & 2 & 0 & 0 & 4 & 0 & 2 & 0 & 0 \\ 2 & 4 & 2 & 1 & 0 & 2 & 1 & 0 & 4 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 2 & 0 \\ 2 & 4 & 2 & 2 & 1 & 0 & 2 & 2 & 0 & 1 & 0 & 4 & 0 & 2 & 0 & 0 & 2 & 0 & 4 & 0 & 0 \\ 2 & 2 & 4 & 1 & 0 & 2 & 2 & 0 & 2 & 0 & 4 & 1 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 4 & 0 \\ 2 & 4 & 2 & 0 & 2 & 1 & 0 & 4 & 2 & 2 & 1 & 0 & 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 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 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 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \end{array}\right)[/math]

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