G := Sym(12);

involutions :=

[
[
    G!(1, 8)(3, 11)(4, 6)(5, 7)(9, 12),
    G!(1, 6)(2, 8)(3, 7)(4, 12)(9, 10),
    G!(1, 12)(2, 11)(3, 10)(4, 8)(5, 6)
]
,
[
    G!(1, 5)(2, 9)(4, 11)(6, 7)(8, 10),
    G!(2, 4)(3, 10)(5, 9)(7, 12)(8, 11),
    G!(1, 7)(2, 5)(3, 9)(6, 11)(10, 12)
]
,
[
    G!(1, 10)(2, 7)(3, 4)(8, 11)(9, 12),
    G!(1, 3)(4, 7)(5, 6)(9, 11)(10, 12),
    G!(1, 4)(2, 8)(3, 11)(5, 10)(6, 7)
]
,
[
    G!(1, 12)(3, 10)(4, 6)(5, 7)(8, 9),
    G!(1, 8)(2, 6)(3, 11)(4, 12)(9, 10),
    G!(1, 6)(2, 11)(3, 7)(4, 8)(5, 12)
]
,
[
    G!(1, 2)(3, 5)(4, 6)(8, 9)(10, 12),
    G!(1, 5)(2, 6)(3, 9)(4, 12)(8, 11),
    G!(1, 9)(2, 3)(4, 8)(5, 12)(6, 7)
]
,
[
    G!(1, 5)(2, 4)(3, 12)(7, 8)(10, 11),
    G!(1, 6)(2, 4)(3, 9)(7, 10)(11, 12),
    G!(1, 5)(3, 9)(4, 8)(6, 10)(7, 11)
]
,
[
    G!(1, 2)(3, 8)(4, 10)(5, 6)(9, 11),
    G!(1, 5)(2, 7)(3, 6)(4, 11)(9, 12),
    G!(1, 9)(2, 8)(3, 12)(4, 7)(5, 10)
]

];

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;