We return to the hexagon presentation for Fi_{22}, and join it just as the double coset enumeration is finishing:
gap> InfoDCE1 := Ignore;
function (...) internal; end
gap> u := DCE(F22Pres);
<< Double coset table "No name" early-closed 24 double 3510 single >>
gap> InfoDCE1 := DCEInfoPrint;;
gap> DCEHOrbits(u);
#I Completed preliminaries, index of M is 7
#I Annotated table
#I Completed orbit 1 size 1
#I Completed orbit 2 size 2816
#I Completed orbit 3 size 693
gap> u.orbsizes;
[ 1, 2816, 693 ]
gap> DCEColAdj(u);
#I Added contribution from 1 part 1
#I Added contribution from 1 part 2
#I Added contribution from 2 part 1
#I Added contribution from 2 part 5
#I Added contribution from 3 part 1
#I Added contribution from 4 part 1
hspace35mm ldots
#I Added contribution from 70 part 1
#I Added contribution from 70 part 3
#I Added contribution from 70 part 4
#I Added contribution from 70 part 5
#I Added contribution from 70 part 7
#I Added contribution from 77 part 1
#I Added contribution from 77 part 5
[ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
[ [ 0, 2816, 0 ], [ 1, 2248, 567 ], [ 0, 2304, 512 ] ],
[ [ 0, 0, 693 ], [ 0, 567, 126 ], [ 1, 512, 180 ] ] ]
gap> DCEColAdjSingle(u,3);
#I Added contribution from 2 part 5
#I Added contribution from 6 part 5
#I Added contribution from 7 part 3
#I Added contribution from 11 part 2
#I Added contribution from 13 part 5
#I Added contribution from 15 part 4
#I Added contribution from 19 part 1
#I Added contribution from 20 part 5
#I Added contribution from 22 part 4
#I Added contribution from 23 part 4
#I Added contribution from 26 part 1
#I Added contribution from 36 part 3
#I Added contribution from 42 part 7
#I Added contribution from 44 part 7
#I Added contribution from 61 part 1
#I Added contribution from 65 part 7
#I Added contribution from 70 part 5
[ [ 0, 0, 693 ], [ 0, 567, 126 ], [ 1, 512, 180 ] ]
GAP 3.4.4