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

involutions :=

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

];

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;