┌───────┐   GAP 4.11.0 of 29-Feb-2020
 │  GAP  │   https://www.gap-system.org
 └───────┘   Architecture: x86_64-apple-darwin16.7.0-default64-kv7
 Configuration:  gmp 6.1.2, GASMAN
 Loading the library and packages ...

 Loading gaprc.
 Packages:   AClib 1.3.2, Alnuth 3.1.2, AtlasRep 2.1.0, AutoDoc 2019.09.04, 
             AutPGrp 1.10.2, Browse 1.8.8, CaratInterface 2.3.3, CRISP 1.4.5, 
             Cryst 4.1.23, CrystCat 1.1.9, CTblLib 1.2.2, FactInt 1.6.3, 
             FGA 1.4.0, Forms 1.2.5, GAPDoc 1.6.3, genss 1.6.6, IO 4.7.0, 
             IRREDSOL 1.4, LAGUNA 3.9.3, orb 4.8.3, Polenta 1.3.9, 
             Polycyclic 2.15.1, PrimGrp 3.4.0, RadiRoot 2.8, recog 1.3.2, 
             ResClasses 4.7.2, SmallGrp 1.4.1, Sophus 1.24, SpinSym 1.5.2, 
             TomLib 1.2.9, TransGrp 2.0.5, utils 0.69, XGAP 4.30
 Try '??help' for help. See also '?copyright', '?cite' and '?authors'
gap> # Numbering is as in Björner and Lutz "Simplicial manifolds, bistellar flips, 
gap> #  and a 16-vertex triangulation of the Poincaré homology 3-sphere"
gap> Reps:=[ [1,3,71,81], [1, 62, 71, 81], [2, 4, 7, 72], [2, 4, 7, 71], [17, 62, 71, 81] ];
[ [ 1, 3, 71, 81 ], [ 1, 62, 71, 81 ], [ 2, 4, 7, 72 ], [ 2, 4, 7, 71 ], 
  [ 17, 62, 71, 81 ] ]
gap> g:=(1,5,2,4,3)(6,11,9,10,8)(61, 111, 92, 102, 82)(62, 112, 91, 101, 81);
(1,5,2,4,3)(6,11,9,10,8)(61,111,92,102,82)(62,112,91,101,81)
gap> h:=(1,3,2)(10,7,9)(101, 71, 91)(102, 72, 92)(6,8,11)(61,81,112)(62,82,111);
(1,3,2)(6,8,11)(7,9,10)(61,81,112)(62,82,111)(71,91,101)(72,92,102)
gap> G:=Group(g,h);
<permutation group with 2 generators>
gap> M:=Union(List(Reps, x->Orbit(G, x, OnSets)));;

gap> # Compute f-vector
gap> List([0..4], x->Size(Union(List(M, y->Combinations(y, x)))));
[ 1, 24, 154, 260, 130 ]

gap> # Verify that every ridge is in exactly 2 facets
gap> Set(Union(List(M, y->Combinations(y,3))), y->Number(M, x->IsSubset(x,y)));
[ 2 ]

gap> # List of facets of the symmetric 24 vertex triangulation of Poincare Homology sphere
gap> M;
[ [ 1, 2, 6, 61 ], [ 1, 2, 6, 62 ], [ 1, 2, 8, 81 ], [ 1, 2, 8, 82 ], 
  [ 1, 2, 10, 101 ], [ 1, 2, 10, 102 ], [ 1, 2, 61, 101 ], [ 1, 2, 62, 81 ], 
  [ 1, 2, 82, 102 ], [ 1, 3, 7, 71 ], [ 1, 3, 7, 72 ], [ 1, 3, 8, 81 ], 
  [ 1, 3, 8, 82 ], [ 1, 3, 11, 111 ], [ 1, 3, 11, 112 ], [ 1, 3, 71, 81 ], 
  [ 1, 3, 72, 111 ], [ 1, 3, 82, 112 ], [ 1, 4, 9, 91 ], [ 1, 4, 9, 92 ], 
  [ 1, 4, 10, 101 ], [ 1, 4, 10, 102 ], [ 1, 4, 11, 111 ], [ 1, 4, 11, 112 ], 
  [ 1, 4, 91, 101 ], [ 1, 4, 92, 111 ], [ 1, 4, 102, 112 ], [ 1, 5, 6, 61 ], 
  [ 1, 5, 6, 62 ], [ 1, 5, 7, 71 ], [ 1, 5, 7, 72 ], [ 1, 5, 9, 91 ], 
  [ 1, 5, 9, 92 ], [ 1, 5, 61, 91 ], [ 1, 5, 62, 71 ], [ 1, 5, 72, 92 ], 
  [ 1, 61, 91, 101 ], [ 1, 62, 71, 81 ], [ 1, 72, 92, 111 ], 
  [ 1, 82, 102, 112 ], [ 2, 3, 6, 61 ], [ 2, 3, 6, 62 ], [ 2, 3, 9, 91 ], 
  [ 2, 3, 9, 92 ], [ 2, 3, 11, 111 ], [ 2, 3, 11, 112 ], [ 2, 3, 61, 111 ], 
  [ 2, 3, 62, 92 ], [ 2, 3, 91, 112 ], [ 2, 4, 7, 71 ], [ 2, 4, 7, 72 ], 
  [ 2, 4, 8, 81 ], [ 2, 4, 8, 82 ], [ 2, 4, 9, 91 ], [ 2, 4, 9, 92 ], 
  [ 2, 4, 71, 91 ], [ 2, 4, 72, 82 ], [ 2, 4, 81, 92 ], [ 2, 5, 7, 71 ], 
  [ 2, 5, 7, 72 ], [ 2, 5, 10, 101 ], [ 2, 5, 10, 102 ], [ 2, 5, 11, 111 ], 
  [ 2, 5, 11, 112 ], [ 2, 5, 71, 112 ], [ 2, 5, 72, 102 ], [ 2, 5, 101, 111 ],
  [ 2, 61, 101, 111 ], [ 2, 62, 81, 92 ], [ 2, 71, 91, 112 ], 
  [ 2, 72, 82, 102 ], [ 3, 4, 6, 61 ], [ 3, 4, 6, 62 ], [ 3, 4, 7, 71 ], 
  [ 3, 4, 7, 72 ], [ 3, 4, 10, 101 ], [ 3, 4, 10, 102 ], [ 3, 4, 61, 72 ], 
  [ 3, 4, 62, 102 ], [ 3, 4, 71, 101 ], [ 3, 5, 8, 81 ], [ 3, 5, 8, 82 ], 
  [ 3, 5, 9, 91 ], [ 3, 5, 9, 92 ], [ 3, 5, 10, 101 ], [ 3, 5, 10, 102 ], 
  [ 3, 5, 81, 101 ], [ 3, 5, 82, 91 ], [ 3, 5, 92, 102 ], [ 3, 61, 72, 111 ], 
  [ 3, 62, 92, 102 ], [ 3, 71, 81, 101 ], [ 3, 82, 91, 112 ], [ 4, 5, 6, 61 ],
  [ 4, 5, 6, 62 ], [ 4, 5, 8, 81 ], [ 4, 5, 8, 82 ], [ 4, 5, 11, 111 ], 
  [ 4, 5, 11, 112 ], [ 4, 5, 61, 82 ], [ 4, 5, 62, 112 ], [ 4, 5, 81, 111 ], 
  [ 4, 61, 72, 82 ], [ 4, 62, 102, 112 ], [ 4, 71, 91, 101 ], 
  [ 4, 81, 92, 111 ], [ 5, 61, 82, 91 ], [ 5, 62, 71, 112 ], 
  [ 5, 72, 92, 102 ], [ 5, 81, 101, 111 ], [ 17, 61, 72, 82 ], 
  [ 17, 61, 72, 111 ], [ 17, 61, 82, 91 ], [ 17, 61, 91, 101 ], 
  [ 17, 61, 101, 111 ], [ 17, 62, 71, 81 ], [ 17, 62, 71, 112 ], 
  [ 17, 62, 81, 92 ], [ 17, 62, 92, 102 ], [ 17, 62, 102, 112 ], 
  [ 17, 71, 81, 101 ], [ 17, 71, 91, 101 ], [ 17, 71, 91, 112 ], 
  [ 17, 72, 82, 102 ], [ 17, 72, 92, 102 ], [ 17, 72, 92, 111 ], 
  [ 17, 81, 92, 111 ], [ 17, 81, 101, 111 ], [ 17, 82, 91, 112 ], 
  [ 17, 82, 102, 112 ] ]
gap>