G := sub<Sym(80)|(2, 4)(5, 9)(6, 11)(8, 14)(12, 21)(17, 24)(18, 28)(20, 30)(22, 32)(25, 35)(26, 31)(27, 37)(29, 41)(33, 43)(34, 47)(36, 46)(39, 49)(42, 58)(51, 
    62)(52, 63)(54, 68)(55, 66)(56, 70)(59, 72)(61, 73)(65, 74)(67, 71)(76, 78),
(1, 2)(3, 6)(4, 7)(5, 10)(8, 15)(9, 16)(11, 19)(12, 18)(13, 22)(14, 23)(17, 27)(20, 31)(21, 25)(24, 34)(26, 36)(28, 39)(29, 42)(30, 43)(32, 
    45)(33, 46)(35, 49)(37, 52)(38, 51)(40, 55)(41, 56)(44, 59)(47, 63)(48, 62)(50, 66)(53, 68)(54, 69)(57, 65)(58, 61)(60, 71)(64, 74)(67, 
    75)(70, 73)(72, 77)(76, 79)(78, 80),
(1, 3)(2, 5)(4, 8)(6, 12)(7, 13)(9, 17)(10, 18)(11, 20)(14, 24)(15, 25)(16, 26)(19, 29)(21, 22)(23, 33)(27, 38)(28, 40)(30, 32)(31, 44)(34, 
    48)(35, 50)(36, 51)(37, 53)(39, 54)(41, 57)(42, 59)(43, 60)(45, 61)(46, 62)(47, 64)(49, 65)(52, 67)(55, 70)(56, 66)(58, 71)(63, 72)(68, 
    76)(69, 73)(74, 78)(75, 79)(77, 80)>;

involutions :=
[
[
G!(2, 4)(5, 9)(6, 11)(8, 14)(12, 21)(17, 24)(18, 28)(20, 30)(22, 32)(25, 35)(26, 31)(27, 37)(29, 41)(33, 43)(34, 47)(36, 46)(39, 49)(42, 58)(51, 
    62)(52, 63)(54, 68)(55, 66)(56, 70)(59, 72)(61, 73)(65, 74)(67, 71)(76, 78),
G!(1, 2)(3, 6)(4, 7)(5, 10)(8, 15)(9, 16)(11, 19)(12, 18)(13, 22)(14, 23)(17, 27)(20, 31)(21, 25)(24, 34)(26, 36)(28, 39)(29, 42)(30, 43)(32, 
    45)(33, 46)(35, 49)(37, 52)(38, 51)(40, 55)(41, 56)(44, 59)(47, 63)(48, 62)(50, 66)(53, 68)(54, 69)(57, 65)(58, 61)(60, 71)(64, 74)(67, 
    75)(70, 73)(72, 77)(76, 79)(78, 80),
G!(1, 3)(2, 5)(4, 8)(6, 12)(7, 13)(9, 17)(10, 18)(11, 20)(14, 24)(15, 25)(16, 26)(19, 29)(21, 22)(23, 33)(27, 38)(28, 40)(30, 32)(31, 44)(34, 
    48)(35, 50)(36, 51)(37, 53)(39, 54)(41, 57)(42, 59)(43, 60)(45, 61)(46, 62)(47, 64)(49, 65)(52, 67)(55, 70)(56, 66)(58, 71)(63, 72)(68, 
    76)(69, 73)(74, 78)(75, 79)(77, 80)
]

];

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;