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


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

// 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, 11846016, 2040025, 71, 34174478, 1990578, 59402955, 23566721, 5921191, 157, 26906044, 21273018, 14064572, 200, 32477765, 3052243, 1688433, 84707286, 3647972, 9384514, 7213632, 1257206, 148546, 134652679, 19531029, 169379, 7158193, 22239, 10157, 716107, 81321416, 28740118, 81684, 4517906, 13672, 14968809, 2812343, 272197, 892131, 45425, 184105162, 7473336, 5000726, 2890324, 1324158, 166400, 539248, 331362, 2286155, 73918681, 2939367, 12737141, 489955, 240196332, 46610954, 23377576, 6001686, 3689936, 673300, 891168, 298772, 8342, 257360557, 43704891, 4057241, 9285751, 136485, 1434215, 1062025, 352323, 3681]); a,b,c,d,e,f,g,h,i,j,k := Explode([GPC.1, GPC.2, GPC.4, GPC.7, GPC.8, GPC.9, GPC.10, GPC.11, GPC.12, GPC.13, GPC.14]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "c4", "d", "e", "f", "g", "h", "i", "j", "k"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 40440);
TransitiveGroup(36, 40601);

// 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