// Magma code for working with abstract group 472392.sf.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([13, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 384696, 385893, 66, 16224938, 16914771, 3781144, 33881, 146, 23534164, 12448037, 2218740, 568676, 44933, 3788010, 1981075, 660860, 330231, 304, 39318, 196579, 3321, 19184263, 13601972, 10024593, 3452014, 813443, 354816, 410, 454904, 37955, 3088809, 21481222, 1397015, 552288, 55051, 4780, 5003866, 31274123, 14287115, 23368200, 10993357, 2100434, 367911, 644512, 84342, 65451684, 15149185, 6077954, 772719, 1125604, 287546, 100489]); a,b,c,d,e,f,g,h,i := Explode([GPC.1, GPC.2, GPC.4, GPC.6, GPC.8, GPC.10, GPC.11, GPC.12, GPC.13]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "d", "d3", "e", "e3", "f", "g", "h", "i"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 30273);

// 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