// Magma code for working with abstract group 1417176.qc.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := PermutationGroup< 36 | (1,6,36)(2,4,34)(3,5,35)(7,8,9)(10,26,30,12,25,29,11,27,28)(13,17,22,14,18,23,15,16,24)(31,33,32), (1,20,29,14,7,16,2,21,28,13,9,17,3,19,30,15,8,18)(4,27,32)(5,25,33,6,26,31)(10,35,12,36,11,34)(22,24) >;

// Order of the group: 
Order(G);

// Exponent of the group: 
Exponent(G);

// Automorphism group: 
AutomorphismGroup(G);

// Composition factors of the group: 
CompositionFactors(G);

// Nilpotency class of the group: 
NilpotencyClass(G);

// Derived length of the group: 
DerivedLength(G);

// Determine if the group G is abelian: 
IsAbelian(G);

// Determine if the group G is cyclic: 
IsCyclic(G);

// Determine if the group G is elementary abelian: 
IsElementaryAbelian(G);

// Determine if the group G is nilpotent: 
IsNilpotent(G);

// Determine if the group G is perfect: 
IsPerfect(G);

// Determine if the group G is simple: 
IsSimple(G);

// Determine if the group G is solvable: 
IsSolvable(G);

// Compute statistics for the group G: 
// Magma code to output the first two rows of the group statistics table
element_orders := [Order(g) : g in G];
orders := Set(element_orders);
printf "Orders: %o\n", orders;
printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G);
cc_orders := [cc[1] : cc in ConjugacyClasses(G)];
printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;

// List of conjugacy classes of the group: 
ConjugacyClasses(G);  // Output not guaranteed to exactly match the LMFDB table

// Compute statistics about the characters of G: 
// Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i
CharacterDegrees(G);

// Define the group with the given generators and relations: 
GPC := PCGroup([14, 3, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 8815464, 10480009, 71, 44716142, 8347026, 4676619, 398849, 8298055, 157, 65129404, 5885898, 5322692, 200, 111712613, 2562355, 14942625, 113552214, 11444852, 16070074, 5834184, 191162, 117516679, 35102613, 20672099, 6265777, 929439, 871997, 291739, 441, 4898888, 1632982, 81684, 122522, 13672, 23688009, 16027223, 10568917, 1234851, 3447425, 933739, 641433, 103427, 555, 5388778, 1097736, 898166, 33316, 149754, 173746955, 52907929, 2939367, 9307925, 489955, 111462636, 39508586, 20815744, 4638870, 3203996, 59078, 23070, 17908141, 3429243, 30862985, 11430775, 5143893]); a,b,c,d,e,f,g,h,i := Explode([GPC.1, GPC.2, GPC.4, GPC.7, GPC.8, GPC.10, GPC.12, GPC.13, GPC.14]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "c4", "d", "e", "e3", "f", "f3", "g", "h", "i"]);

// Define the group as a permutation group: 
PermutationGroup< 36 | (1,6,36)(2,4,34)(3,5,35)(7,8,9)(10,26,30,12,25,29,11,27,28)(13,17,22,14,18,23,15,16,24)(31,33,32), (1,20,29,14,7,16,2,21,28,13,9,17,3,19,30,15,8,18)(4,27,32)(5,25,33,6,26,31)(10,35,12,36,11,34)(22,24) >;

// Define the group from the transitive group database: 
TransitiveGroup(36, 40441);

// The primary decomposition of the group: 
PrimaryInvariants(G);

// The abelianization of the group: 
quo< G | CommutatorSubgroup(G) >;

// List of subgroups of the group: 
Subgroups(G);

// Center of the group: 
Center(G);

// Commutator subgroup of the group G: 
CommutatorSubgroup(G);

// Frattini subgroup of the group G: 
FrattiniSubgroup(G);

// Fitting subgroup of the group G: 
FittingSubgroup(G);

// Radical of the group G: 
Radical(G);

// Socle of the group G: 
Socle(G);

// Derived series of the group G: 
DerivedSeries(G);

// Chief series of the group G: 
ChiefSeries(G);

// The lower central series of the group G: 
LowerCentralSeries(G);

// The upper central series of the group G: 
UpperCentralSeries(G);

// Character table: 
CharacterTable(G);  // Output not guaranteed to exactly match the LMFDB table