G := PGL(2,64);

involutions :=

[
[
    G!(1, 36)(2, 17)(3, 49)(4, 40)(5, 37)(6, 62)(7, 23)(8, 24)(9, 20)(10, 44)(11, 
        22)(12, 50)(13, 15)(14, 29)(16, 18)(19, 46)(21, 52)(25, 34)(26, 59)(27, 
        56)(28, 47)(30, 60)(31, 35)(32, 54)(33, 39)(38, 43)(41, 51)(42, 64)(45, 
        55)(53, 58)(57, 63)(61, 65),
    G!(1, 7)(2, 17)(3, 53)(4, 63)(5, 23)(6, 10)(8, 42)(9, 54)(11, 37)(12, 29)(13, 
        27)(14, 24)(15, 40)(16, 61)(18, 26)(19, 49)(20, 22)(21, 51)(25, 36)(28, 
        62)(30, 55)(31, 38)(32, 39)(33, 59)(34, 45)(41, 58)(43, 57)(44, 52)(46, 
        56)(47, 65)(48, 50)(60, 64),
    G!(1, 6)(2, 51)(3, 18)(4, 36)(5, 53)(7, 21)(8, 17)(9, 65)(10, 61)(11, 16)(12, 
        29)(13, 46)(14, 64)(15, 31)(19, 54)(20, 32)(22, 40)(23, 43)(24, 30)(25, 
        38)(26, 49)(27, 37)(28, 63)(33, 56)(34, 35)(39, 59)(42, 60)(44, 57)(45, 
        55)(47, 48)(50, 62)(52, 58)
]
,
[
    G!(1, 34)(2, 46)(3, 15)(5, 58)(6, 27)(7, 39)(8, 16)(9, 61)(10, 45)(11, 36)(12,
        64)(13, 30)(14, 25)(17, 35)(18, 33)(19, 41)(20, 44)(21, 50)(22, 31)(23, 
        52)(24, 26)(28, 63)(29, 53)(32, 54)(37, 62)(38, 56)(40, 55)(42, 51)(43, 
        60)(47, 49)(48, 59)(57, 65),
    G!(1, 51)(2, 4)(3, 8)(5, 17)(6, 33)(7, 11)(9, 53)(10, 28)(12, 49)(13, 40)(14, 
        22)(15, 57)(16, 25)(18, 36)(19, 48)(20, 59)(21, 30)(24, 32)(26, 52)(27, 
        63)(29, 41)(31, 54)(34, 62)(35, 39)(37, 58)(38, 43)(42, 44)(45, 60)(46, 
        56)(47, 61)(50, 64)(55, 65),
    G!(1, 50)(2, 42)(3, 5)(4, 38)(6, 11)(7, 20)(8, 62)(9, 14)(12, 23)(13, 65)(15, 
        17)(16, 47)(18, 43)(19, 35)(21, 22)(24, 56)(25, 51)(26, 49)(27, 39)(28, 
        32)(29, 61)(30, 37)(31, 40)(33, 41)(34, 60)(36, 59)(44, 52)(45, 54)(46, 
        58)(48, 55)(53, 57)(63, 64)
]

];

geometries := [* *];


for i in [1..#involutions] do
	G1 := sub<G|involutions[i][2],involutions[i][3]>;
	G2 := sub<G|involutions[i][1],involutions[i][3]>;
	G3 := sub<G|involutions[i][1],involutions[i][2]>;
	Append(~geometries,CosetGeometry(G,{G1,G2,G3}));
end for;