// Magma code for working with abstract group 78732.je.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([11, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 22, 641532, 2271260, 342850, 90, 1570803, 705950, 2880904, 22785, 27416, 1774877, 1102480, 263367, 275456, 10939, 205134, 1287611, 262906, 322746, 13217, 2371, 3465799, 1740834, 841133, 26990, 9049, 2598164, 2328489, 887060, 1236871, 270972, 44603, 5025, 7346, 352846]); a,b,c,d,e,f,g,h,i := Explode([GPC.1, GPC.3, GPC.5, GPC.6, GPC.7, GPC.8, GPC.9, GPC.10, GPC.11]); AssignNames(~GPC, ["a", "a2", "b", "b2", "c", "d", "e", "f", "g", "h", "i"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 18899);

// 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