# MAPLE. April 28, 2009.
#
# Higman triality scheme from U_6(2) (3 fibres)
d := 4;
v := [1 ,    224  ,   567 ,   2592,   840];
verts := 4224;

Q := matrix(5,5,[
 1,    252 ,   3465,    504 ,    2,
 1,    54  ,      0,    -54 ,   -1,
 1,   52/3 ,    -55,   104/3,    2,
 1,   -14/3,      0,   14/3 ,   -1,
 1,    -12 ,     33,    -24 ,    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 ,   252 ,     0  ,     0  ,   0],
[1 ,  148/3,   605/3,     0  ,   0],
[0 ,  44/3 ,    208 ,   88/3 ,   0],
[0 ,    0  ,   605/3,   148/3,   1],
[0 ,    0  ,     0  ,    252 ,   0]
]);

        [0    224     0      0     0]
        [                           ]
        [1     31    81     81    30]
        [                           ]
 L_1 := [0     32     0    192     0]
        [                           ]
        [0      7    42    105    70]
        [                           ]
        [0      8     0    216     0]

        [0     0    567      0      0]
        [                            ]
        [0    81      0    486      0]
        [                            ]
 L_2 := [1     0    246      0    320]
        [                            ]
        [0    42      0    525      0]
        [                            ]
        [0     0    216      0    351]

        [0      0      0    2592      0]
        [                              ]
        [0     81    486    1215    810]
        [                              ]
 L_3 := [0    192      0    2400      0]
        [                              ]
        [1    105    525    1191    770]
        [                              ]
        [0    216      0    2376      0]

        [0     0      0      0    840]
        [                            ]
        [0    30      0    810      0]
        [                            ]
 L_4 := [0     0    320      0    520]
        [                            ]
        [0    70      0    770      0]
        [                            ]
        [1     0    351      0    488]

        [1     224    567     2592    840]
        [                                ]
        [1      48     39      -48    -40]
        [                                ]
  P :=  [1       0     -9        0      8]
        [                                ]
        [1     -24     39       24    -40]
        [                                ]
        [1    -112    567    -1296    840]

        [1     252     3465     504      2]
        [                                 ]
        [1     54         0     -54     -1]
        [                                 ]
  Q :=  [1    52/3      -55    104/3     2]
        [                                 ]
        [1    -14/3       0    14/3     -1]
        [                                 ]
        [1     -12       33     -24      2]

        [0     252       0        0      0]
        [                                 ]
        [1    148/3    605/3      0      0]
        [                                 ]
 Ls1 := [0    44/3      208     88/3     0]
        [                                 ]
        [0      0      605/3    148/3    1]
        [                                 ]
        [0      0        0       252     0]

        [0      0      3465      0       0]
        [                                 ]
        [0    605/3    2860    1210/3    0]
        [                                 ]
 Ls2 := [1     208     2838     416      2]
        [                                 ]
        [0    605/3    2860    1210/3    0]
        [                                 ]
        [0      0      3465      0       0]

        [0      0        0        504     0]
        [                                  ]
        [0      0      1210/3    296/3    2]
        [                                  ]
 Ls3 := [0    88/3      416      176/3    0]
        [                                  ]
        [1    148/3    1210/3    148/3    1]
        [                                  ]
        [0     252       0        252     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]