G := sub<Sym(24)|(1, 22)(3, 7)(5, 19)(6, 21)(9, 12)(10, 18)(13, 24)(14, 17),(1, 24)(2, 7)(5, 16)(8, 13)(9, 17)(11, 20)(14, 18)(15, 22),(1, 13)(3, 23)(4, 17)(5, 20)(6, 8)(7, 21)(10, 24)(15, 16)>;

involutions :=

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

];

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;