S_4: the six subgroups have the following lists of generators: [ [ w5, w1 * w2 * w3 * w2 * w1 * w5, w4 * w5, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ], [ w2 * w3 * w2 * w4 * w3 * w5 * w3 * w2 * w4 * w3 * w5, w1 * w2 * w3 * w2 * w1 * w5, w4 * w5, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ], [ w2 * w1 * w3 * w2 * w1 * w5 * w3 * w2, w3 * w5 * w3 * w2, w3 * w4 * w5 * w3, w2 * w3 * w4 * w5 * w3 * w2 ], [ w3 * w4 * w5 * w3 * w4, w1 * w2 * w3 * w2 * w1 * w5, w4 * w5, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ], [ w2 * w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1, w1 * w2 * w3 * w2 * w1 * w4, w2 * w4 * w1 * w2, w4 * w1 * w2 * w3 * w2 * w1 * w4 * w3 ], [ w5, w1 * w2 * w3 * w2 * w1 * w5, w2 * w1 * w2 * w5, w1 * w2 * w3 * w2 * w5 * w3 * w2 * w1 ] ] Only the second one has nontrivial cohomology. The orbit sizes are respectively [4,4,4,4] [8,8] [4,4,8] [8,8] [2,6,8] [1,1,4,4,6] so unless the orbit sizes are [8,8], there is no cohomology. In these cases, consider the stabilizer of line 1. This group also fixes lines 6, 7, and 15. However, for the second group, lines 1 and 15 are conjugate over Q, while lines 1 and 6 are conjugate over Q for the fourth group. Similarly if one considers the stabilizer of line 2 instead: it also fixes 3, 10, and 16, while 2 and 3 are in the same orbit for the first group and 2 and 16 for the second. Thus there is nontrivial cohomology if and only if the orbits are of sizes [8,8] and, over the field of definition of one line, two rational lines that intersect were not originally conjugate. (Z/2)^2 x D_4: the 2 such subgroups have generators [ [ w5, w4 * w5, w3 * w4 * w5 * w3, w2 * w3 * w4 * w5 * w3 * w2, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ], [ w5, w2 * w1 * w2 * w5, w4 * w5, w2 * w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w5^-1, w2 * w3 * w4 * w5 * w3 * w2 ] ] Only the second has nontrivial cohomology. The orbit sizes are respectively [16] [8,8] so the cohomology is nontrivial if and only if there is more than one orbit of lines. (Z/2)^2 x A_4: the subgroups are [ [ w2 * w3 * w2 * w5, w4 * w5, w3 * w4 * w5 * w3, w2 * w3 * w4 * w5 * w3 * w2, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ], [ w1 * w2, w5, w4 * w5, w2 * w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w5^-1, w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w5^-1 ] ] Only the second has nontrivial cohomology. The orbit sizes are respectively [16] [8,8] so the cohomology is nontrivial if and only if there is more than one orbit of lines. Z/2 x S_4: there are 8 subgroups, which are [ [ w5, w1 * w2 * w3 * w2 * w1 * w5, w4 * w5, w2 * w3 * w2 * w4 * w3 * w5 * w3 * w2 * w4 * w3, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ], [ w2 * w3 * w4 * w5 * w3 * w2 * w4, w1 * w2 * w3 * w2 * w1 * w5, w4 * w5, w2 * w3 * w2 * w4 * w3 * w5 * w3 * w2 * w4 * w3, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ], [ w2 * w1 * w3 * w2 * w1 * w5 * w3 * w2, w3 * w5 * w3 * w2, w4 * w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 * w4, w3 * w4 * w5 * w3, w2 * w3 * w4 * w5 * w3 * w2 ], [ w2 * w3 * w4 * w5 * w3 * w2 * w4, w1 * w2 * w3 * w2 * w1 * w5, w4 * w5, w3 * w4 * w5 * w3, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ], [ w5, w1 * w2 * w3 * w2 * w1 * w5, w4 * w5, w3 * w4 * w5 * w3, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ], [ w1, w1 * w2, w5, w2 * w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w5^-1, w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w5^-1 ], [ w4 * w1 * w5, w1 * w2, w5, w2 * w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w5^-1, w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w5^-1 ], [ w5, w1 * w2 * w3 * w2 * w1 * w5, w2 * w1 * w2 * w5, w1 * w2 * w3 * w2 * w5 * w3 * w2 * w1, w2 * w4 * w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 ] ] Of these, the first, third, and seventh have nontrivial cohomology. The orbit sizes are [8,8] [16] [8,8] [16] [8,8] [4,4,8] [8,8] [2,6,8] establishing that if there are not exactly two orbits, then there is no cohomology. The orbit sizes of the stabilizer of one line from either orbit are (considering only groups with two orbits) [1,1,1,1,3,3,3,3] [1,1,2,3,3,6] [1,1,1,1,3,3,3,3] [1,1,2,3,3,6] so if there are exactly two lines defined over the stabilizer of one line, then there is nontrivial cohomology. If there are four, then if we chose the stabilizer of 1 they are 1,6,7,15. For the first group, 1 and 15, which do not intersect, are conjugate; for the third (fifth on the original list) 1 and 6 are, which do. Alternatively, if we chose the other orbit it would be the stabilizer of 14, which fixes 5, 8, 12, 14. For the first group, 12 and 14, which do not intersect, are conjugate; for the second group, 5 and 14, which do. (Z/2)^2 x S_4: there are 2 subgroups, namely [ [ w1, w1 * w2, w5, w4 * w5, w2 * w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w5^-1, w1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w2 * w4 * w2 * w4 * w2 * w1 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w1 * w3 * w1 * w5 * w3^3 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2^2 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w1 * w3 * w1 * w5 * w3 * w2 * w1 * w3 * w1 * w5 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w3^2 * w1 * w5 * w3 * w1 * w5 * w3 * w1 * w3 * w1 * w5 * w2 * w4 * w2 * w1 * w4^-1 * w3^-1 * w5^-1 ], [ w5, w1 * w2 * w3 * w2 * w1 * w5, w4 * w5, w3 * w4 * w5 * w3, w2 * w3 * w4 * w5 * w3 * w2, w1 * w2 * w3 * w4 * w5 * w3 * w2 * w1 ] ] Only the first has nontrivial cohomology. The orbit sizes are [8,8] [16] so the cohomology is nontrivial if and only if there are 2 orbits of lines. ---- These presentations can be remarkably ugly, so maybe it's better to give them as permutation groups, which also makes life easier for anyone who wants to check this (especially the information on orbits). The generators are given in the following order, which coincides with the s_1, ..., s_5 of Manin 25.5.7 when we index the lines in the usual way: the e_i in lexicographic order, then the l_{ij} in lexicographic order, then q. w1 = (1, 2)(7, 10)(8, 11)(9, 12) w2 = (2, 3)(6, 7)(11, 13)(12, 14) w3 = (3, 4)(7, 8)(10, 11)(14, 15) w4 = (1, 10)(2, 7)(3, 6)(15, 16) w5 = (4, 5)(8, 9)(11, 12)(13, 14) Z/2 x A_4 (in the same order as in the paper): [ Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (1, 3, 11)(2, 8, 13)(4, 6, 10)(9, 16, 14) (1, 14)(2, 4)(3, 9)(5, 7)(6, 8)(10, 13)(11, 16)(12, 15) (1, 11)(2, 8)(3, 5)(4, 6)(7, 9)(10, 12)(13, 15)(14, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (2, 3, 4)(6, 7, 8)(10, 13, 11)(12, 14, 15) (2, 13)(3, 11)(4, 10)(9, 16) (1, 14)(2, 4)(3, 9)(5, 7)(6, 8)(10, 13)(11, 16)(12, 15) (1, 15)(2, 3)(4, 9)(5, 8)(6, 7)(10, 16)(11, 13)(12, 14), Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (1, 3, 11)(2, 8, 13)(4, 6, 10)(9, 16, 14) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 11)(2, 8)(3, 5)(4, 6)(7, 9)(10, 12)(13, 15)(14, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (1, 3, 11)(2, 8, 13)(4, 6, 10)(9, 16, 14) (2, 11)(3, 13)(4, 10)(6, 7)(9, 16)(12, 14) (1, 8)(3, 13)(4, 7)(6, 10)(9, 14)(12, 16) (1, 8)(2, 11)(3, 13)(4, 16)(5, 15)(6, 14)(7, 12)(9, 10) ] S_4: [ Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (4, 5)(8, 9)(11, 12)(13, 14) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (1, 15)(2, 3)(4, 8)(5, 9)(6, 7)(10, 16)(11, 14)(12, 13) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (1, 4)(2, 5)(6, 15)(7, 13)(9, 11)(10, 14) (2, 3, 5)(6, 7, 9)(10, 14, 12)(11, 13, 15) (1, 11)(2, 8)(3, 5)(4, 6)(7, 9)(10, 12)(13, 15)(14, 16) (1, 13)(2, 5)(3, 8)(4, 7)(6, 9)(10, 14)(11, 15)(12, 16), Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (1, 11, 10, 12)(2, 8, 7, 9)(3, 5, 6, 4)(13, 16, 14, 15) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (1, 3)(2, 15)(4, 11)(5, 12)(6, 10)(7, 16)(8, 14)(9, 13) (1, 4, 10)(2, 7, 13)(3, 6, 11)(9, 16, 15) (1, 6)(2, 7)(3, 10)(8, 13)(9, 14)(15, 16) (2, 13)(3, 10)(4, 11)(7, 8)(9, 16)(14, 15), Permutation group acting on a set of cardinality 16 Order = 24 = 2^3 * 3 (4, 5)(8, 9)(11, 12)(13, 14) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 3)(4, 5)(6, 10)(8, 14)(9, 13)(11, 12) (1, 5)(3, 4)(6, 12)(7, 15)(8, 14)(10, 11) ] (Z/2)^2 x D_4: [ Permutation group acting on a set of cardinality 16 Order = 32 = 2^5 (4, 5)(8, 9)(11, 12)(13, 14) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 11)(2, 8)(3, 5)(4, 6)(7, 9)(10, 12)(13, 15)(14, 16) (1, 13)(2, 5)(3, 8)(4, 7)(6, 9)(10, 14)(11, 15)(12, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 32 = 2^5 (4, 5)(8, 9)(11, 12)(13, 14) (1, 3)(4, 5)(6, 10)(8, 14)(9, 13)(11, 12) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 3)(2, 15)(4, 12)(5, 11)(6, 10)(7, 16)(8, 13)(9, 14) (1, 13)(2, 5)(3, 8)(4, 7)(6, 9)(10, 14)(11, 15)(12, 16) ] (Z/2)^2 x A_4: [ Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (2, 5, 4)(6, 9, 8)(10, 14, 13)(11, 12, 15) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 11)(2, 8)(3, 5)(4, 6)(7, 9)(10, 12)(13, 15)(14, 16) (1, 13)(2, 5)(3, 8)(4, 7)(6, 9)(10, 14)(11, 15)(12, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (1, 3, 2)(6, 7, 10)(8, 13, 11)(9, 14, 12) (4, 5)(8, 9)(11, 12)(13, 14) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 3)(2, 15)(4, 12)(5, 11)(6, 10)(7, 16)(8, 13)(9, 14) (1, 2)(3, 15)(4, 14)(5, 13)(6, 16)(7, 10)(8, 11)(9, 12) ] Z/2 x S_4: [ Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (4, 5)(8, 9)(11, 12)(13, 14) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 15)(2, 3)(4, 9)(5, 8)(6, 7)(10, 16)(11, 13)(12, 14) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (1, 13, 10, 14)(2, 5, 7, 4)(3, 8, 6, 9)(11, 16, 12, 15) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 15)(2, 3)(4, 9)(5, 8)(6, 7)(10, 16)(11, 13)(12, 14) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (1, 4)(2, 5)(6, 15)(7, 13)(9, 11)(10, 14) (2, 3, 5)(6, 7, 9)(10, 14, 12)(11, 13, 15) (1, 4)(2, 14)(3, 12)(5, 10)(6, 11)(7, 13)(8, 16)(9, 15) (1, 11)(2, 8)(3, 5)(4, 6)(7, 9)(10, 12)(13, 15)(14, 16) (1, 13)(2, 5)(3, 8)(4, 7)(6, 9)(10, 14)(11, 15)(12, 16), Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (1, 13, 10, 14)(2, 5, 7, 4)(3, 8, 6, 9)(11, 16, 12, 15) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 11)(2, 8)(3, 5)(4, 6)(7, 9)(10, 12)(13, 15)(14, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (4, 5)(8, 9)(11, 12)(13, 14) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 11)(2, 8)(3, 5)(4, 6)(7, 9)(10, 12)(13, 15)(14, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16), Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (1, 2)(7, 10)(8, 11)(9, 12) (1, 3, 2)(6, 7, 10)(8, 13, 11)(9, 14, 12) (4, 5)(8, 9)(11, 12)(13, 14) (1, 3)(2, 15)(4, 12)(5, 11)(6, 10)(7, 16)(8, 13)(9, 14) (1, 2)(3, 15)(4, 14)(5, 13)(6, 16)(7, 10)(8, 11)(9, 12), Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (1, 7)(2, 10)(3, 6)(4, 5)(8, 12)(9, 11)(13, 14)(15, 16) (1, 3, 2)(6, 7, 10)(8, 13, 11)(9, 14, 12) (4, 5)(8, 9)(11, 12)(13, 14) (1, 3)(2, 15)(4, 12)(5, 11)(6, 10)(7, 16)(8, 13)(9, 14) (1, 2)(3, 15)(4, 14)(5, 13)(6, 16)(7, 10)(8, 11)(9, 12), Permutation group acting on a set of cardinality 16 Order = 48 = 2^4 * 3 (4, 5)(8, 9)(11, 12)(13, 14) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 3)(4, 5)(6, 10)(8, 14)(9, 13)(11, 12) (1, 5)(3, 4)(6, 12)(7, 15)(8, 14)(10, 11) (1, 6)(2, 16)(3, 10)(4, 11)(5, 12)(7, 15)(8, 14)(9, 13) ] (Z/2)^2 x S_4: [ Permutation group acting on a set of cardinality 16 Order = 96 = 2^5 * 3 (1, 2)(7, 10)(8, 11)(9, 12) (1, 3, 2)(6, 7, 10)(8, 13, 11)(9, 14, 12) (4, 5)(8, 9)(11, 12)(13, 14) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 3)(2, 15)(4, 12)(5, 11)(6, 10)(7, 16)(8, 13)(9, 14) (1, 2)(3, 15)(4, 14)(5, 13)(6, 16)(7, 10)(8, 11)(9, 12), Permutation group acting on a set of cardinality 16 Order = 96 = 2^5 * 3 (4, 5)(8, 9)(11, 12)(13, 14) (1, 5, 4)(6, 12, 11)(7, 14, 13)(8, 9, 15) (1, 10)(2, 7)(3, 6)(4, 5)(8, 9)(11, 12)(13, 14)(15, 16) (1, 11)(2, 8)(3, 5)(4, 6)(7, 9)(10, 12)(13, 15)(14, 16) (1, 13)(2, 5)(3, 8)(4, 7)(6, 9)(10, 14)(11, 15)(12, 16) (1, 5)(2, 13)(3, 11)(4, 10)(6, 12)(7, 14)(8, 15)(9, 16) ]