G := PGL(2,27);

involutions :=

[
[
    G!(2, 19)(3, 26)(4, 21)(5, 7)(6, 12)(8, 15)(9, 27)(10, 23)(11, 17)(13, 28)(14,
        24)(16, 18)(20, 25),
    G!(1, 20)(2, 22)(3, 10)(4, 21)(6, 26)(7, 24)(8, 27)(9, 13)(11, 16)(12, 17)(14,
        28)(15, 19)(18, 25),
    G!(1, 7)(2, 28)(3, 10)(4, 22)(5, 18)(6, 12)(8, 23)(9, 19)(11, 13)(14, 25)(15, 
        20)(16, 27)(21, 26)
]
,
[
    G!(2, 24)(3, 9)(5, 28)(6, 25)(7, 18)(8, 27)(10, 14)(11, 12)(13, 16)(15, 
        26)(17, 20)(19, 23)(21, 22),
    G!(1, 8)(2, 4)(3, 23)(6, 9)(7, 14)(11, 13)(12, 20)(15, 24)(16, 26)(17, 27)(18,
        22)(19, 28)(21, 25),
    G!(1, 16)(2, 19)(3, 25)(4, 18)(5, 11)(6, 17)(7, 20)(8, 10)(9, 21)(13, 22)(14, 
        23)(15, 27)(26, 28)
]

];

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;