# MAPLE. April 28, 2009.
#
# Higman triality scheme from L_3(4) (3 copies of Gewirtz)
d := 4;
v := [1, 40, 45, 72, 10];
verts := 168;

Q := matrix([
[1 ,  20 , 105 ,   40 ,   2],
[1 ,   6 ,   0 ,   -6 ,  -1],
[1 , 4/3 ,  -7 ,  8/3 ,   2],
[1 ,-10/3,   0 , 10/3 ,  -1],
[1 ,  -8 ,  21 ,  -16 ,   2]
]);

#  This tridiagonal  matrix, L_1-star,  allows us to fill out the cols of Q
#  if we are only given the cosines (normalized column 1 of Q).
L := matrix([
[0 ,   20  ,   0    ,   0    ,  0  ],
[1 ,  8/3  , 49/3   ,   0    ,  0  ],
[0 , 28/9  , 32/3   , 56/9   ,  0  ],
[0 ,    0  , 49/3   ,  8/3   ,  1  ],
[0 ,    0  ,   0    ,  20    ,  0  ]
]);

         [0    40     0     0    0]
         [                        ]
         [1    11    18     9    1]
         [                        ]
 L_1 :=  [0    16     0    24    0]
         [                        ]
         [0     5    15    15    5]
         [                        ]
         [0     4     0    36    0]

         [0     0    45     0    0]
         [                        ]
         [0    18     0    27    0]
         [                        ]
 L_2 :=  [1     0    36     0    8]
         [                        ]
         [0    15     0    30    0]
         [                        ]
         [0     0    36     0    9]

         [0     0     0    72    0]
         [                        ]
         [0     9    27    27    9]
         [                        ]
 L_3 :=  [0    24     0    48    0]
         [                        ]
         [1    15    30    21    5]
         [                        ]
         [0    36     0    36    0]

         [0    0    0    0    10]
         [                      ]
         [0    1    0    9     0]
         [                      ]
 L_4 :=  [0    0    8    0     2]
         [                      ]
         [0    5    0    5     0]
         [                      ]
         [1    0    9    0     0]

           [1     40    45     72    10]
           [                           ]
           [1     12     3    -12    -4]
           [                           ]
    P :=   [1      0    -3      0     2]
           [                           ]
           [1     -6     3      6    -4]
           [                           ]
           [1    -20    45    -36    10]

           [1     20      105     40      2]
           [                               ]
           [1      6        0     -6     -1]
           [                               ]
   Q :=    [1     4/3      -7    8/3      2]
           [                               ]
           [1    -10/3      0    10/3    -1]
           [                               ]
           [1     -8       21    -16      2]


         [0     20      0       0      0]
         [                              ]
         [1    8/3     49/3     0      0]
         [                              ]
 Ls1 :=  [0    28/9    32/3    56/9    0]
         [                              ]
         [0     0      49/3    8/3     1]
         [                              ]
         [0     0       0       20     0]

         [0     0      105     0      0]
         [                             ]
         [0    49/3     56    98/3    0]
         [                             ]
 Ls2 :=  [1    32/3     70    64/3    2]
         [                             ]
         [0    49/3     56    98/3    0]
         [                             ]
         [0     0      105     0      0]

         [0     0       0       40      0]
         [                               ]
         [0     0      98/3    16/3     2]
         [                               ]
 Ls3 :=  [0    56/9    64/3    112/9    0]
         [                               ]
         [1    8/3     98/3     8/3     1]
         [                               ]
         [0     20      0       20      0]

         [0    0    0    0    2]
         [                     ]
         [0    0    0    2    0]
         [                     ]
 Ls4 :=  [0    0    2    0    0]
         [                     ]
         [0    1    0    1    0]
         [                     ]
         [1    0    0    0    1]