G := PSL(2,27);

involutions :=

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

];

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;