with(GroupTheory):
G := Group({[[1, 2]], [[3, 4]]});
Orbit(G);
G := Group({[[1, 2], [3, 4]]});
Orbit(G);

GroupTheory:-PermutationGroup(

{module () local cycles, p, d, work; option object; end module,

module () local cycles, p, d, work; option object; end module},

degree = 4)
Error, invalid input: GroupTheory:-Orbit expects its 1st argument, point, to be of type posint, but received module () local labels, minSupp, maxSupp, suppSize, AtkinsonsAlgorithm, IsSimpleGroupOrder, doDerivedSeries, doLowerCentralSeries, Intersection2, RightCosetRepresentatives, LeftCosetRepresentatives, PRA, `Giant?`, `Even?`, doStab1, doStab, CycleIndexMonomial; export generator_list, n, supergroup, Sylows, pCores, ModulePrint, ModuleDeconstruct, Generators, Orbit, Orbits, IsTransitive, Transitivity, IsPrimitive, GroupOrder, Elements, IsAbelian, IsElementary, IsSimple, ConjugacyClass, ConjugacyClasses, CayleyTable, Centre, DerivedSubgrou...
GroupTheory:-PermutationGroup(

{module () local cycles, p, d, work; option object; end module},

degree = 4)
Error, invalid input: GroupTheory:-Orbit expects its 1st argument, point, to be of type posint, but received module () local labels, minSupp, maxSupp, suppSize, AtkinsonsAlgorithm, IsSimpleGroupOrder, doDerivedSeries, doLowerCentralSeries, Intersection2, RightCosetRepresentatives, LeftCosetRepresentatives, PRA, `Giant?`, `Even?`, doStab1, doStab, CycleIndexMonomial; export generator_list, n, supergroup, Sylows, pCores, ModulePrint, ModuleDeconstruct, Generators, Orbit, Orbits, IsTransitive, Transitivity, IsPrimitive, GroupOrder, Elements, IsAbelian, IsElementary, IsSimple, ConjugacyClass, ConjugacyClasses, CayleyTable, Centre, DerivedSubgrou...

how to calculate Ferrari resolvent of x^4-c1*x^3+c2*x^2-c3*x+c4

for the questions below

you can use A,B,C as single varible or matrix form if needed

you can use A,B,C as single varible or matrix form if needed

1.how to calculate AA1, AA2 and B2 in terms of A,B,C?

2.how to calculate A,B,C in terms of AA1, AA2 and B2 ?

xrestart;

with(LinearAlgebra):
A := Matrix([[a1,a2,a3],[a4,a5,a6],[a7,a8,a9]]);
B := Matrix([[b1,b2,b3],[b4,b5,b6],[b7,b8,b9]]);
C := Matrix([[c1,c2,c3],[c4,c5,c6],[c7,c8,c9]]);
#AA1,AA2, C:=seq(Matrix(3, symbol=i), i=[a,b,c]);
#AA1 := Matrix([[aaa1,0,0],[0,aaa1,0],[0,0,aaa1]]);
#AA2 := Matrix([[aab1,0,0],[0,aab1,0],[0,0,aab1]]);
#B2 := Matrix([[aabb1,0,0],[0,aabb1,0],[0,0,aabb1]]);
AA1 := Matrix([[aaa1,aaa2,aaa3],[aaa4,aaa5,aaa6],[aaa7,aaa8,aaa9]]);
AA2 := Matrix([[aab1,aab2,aab3],[aab4,aab5,aab6],[aab7,aab8,aab9]]);
B2 := Matrix([[aabb1,aabb2,aabb3],[aabb4,aabb5,aabb6],[aabb7,aabb8,aabb9]]);
eq2 := C.A+C.B;
eq3 := C.B+C;
eq4 := B+C.A;
ABC := fsolve(map(t->Equate(op(t))[], [eq2=AA1,eq3=AA2,eq4=B2]));

do not know why Basis got error in this case.

how to calculate this Basis?

with(LinearAlgebra):
prej := Matrix([[diff(eq2,a),diff(eq2,b),diff(eq2,c)],[diff(eq3,a),diff(eq3,b),diff(eq3,c)],[diff(eq4,a),diff(eq4,b),diff(eq4,c)]]);
jaco := Determinant(prej);
jaco := -a*b*c^2+c^2;
g3 := [diff(jaco,a),diff(jaco,b),diff(jaco,c)];
K := [r-g3[1],u-g3[2],v-g3[3]];
G := Basis(K, 'tord', degrevlex(r,u,v));

Error, (in LinearAlgebra:-Basis) invalid input: LinearAlgebra:-Basis expects its 1st argument, V, to be of type {Vector, {list(Vector), set(Vector)}} but received [b*c^2+r, a*c^2+u, 2*a*b*c-2*c+v]

i find that Suzuki group is very large, 

and finding subrgoup is still not solved 

and when find subgroup of suzuki group that is very slow

i would like to find two subgroup which are not isomorphric and then

prove a normaliser that make them isomorphric

with(GroupTheory):
G := SuzukiGroup();
IsFinite(G);
gorder := GroupOrder(G);
ifactor(gorder);
spg := SylowSubgroup(2, G);
IsAbelian(spg);
Elements(spg);
lprint(%);
H := Subgroup(Elements(G), spg);
N := Normaliser(G, spg);
#N := Normaliser(spg, G);
Elements(N);
lprint(%);
Elements(G);
H2 := Subgroup({[[5,2],[3,4]]}, G);
H2 := Subgroup(Elements(G), G);
elements2 := convert(Elements(G), 'list');
generators := map(ListTools:-Search, [Perm([[1,2,3]])], elements2);
H2 := Subgroup([52], G);
SubgroupMembership(H2, G);
H2 := Subgroup([3], G);
elist := Elements(H2);
AreConjugate(elist[2], elist[3], N); #N*elist[1]*N^(-1) = elist[2]
for ii from 1 to nops(elist) do
for jj from ii+1 to nops(elist) do
if AreIsomorphic(Group(elist[ii]),Group(elist[jj])) = false then
print("noniso");
print(ii);
print(jj);
end if:
od:
od:
AreConjugate(elist[8], elist[22], N); #N*elist[1]*N^(-1) = elist[2]