// Magma code for working with abstract group 5668704.it.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([16, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 1825344, 25720065, 81, 16099586, 32752546, 312656643, 172391827, 80112611, 179, 353990404, 102518420, 36812676, 310394309, 184733589, 51572197, 10866293, 10102245, 277, 315366918, 167957014, 82305254, 11097462, 22454950, 531383815, 188587031, 168086055, 46609975, 14559047, 3263319, 4905703, 375, 754610696, 393341208, 186664, 26645816, 32757768, 7331992, 2940296, 448039689, 138265, 241961, 55388217, 36409033, 4154969, 5834985, 2022841, 506057, 398107786, 215322650, 494250, 40335034, 10175690, 4614810, 6424810, 1029722, 557706, 737372171, 128480283, 142207531, 28422203, 37539147, 14093659, 2268411, 1171723, 219611, 73323, 446953740, 7637788, 287853740, 44883132, 24261196, 5813276, 4346892, 235996, 1331132, 27020, 674666509, 47029309, 7838301, 1306493, 587911694, 576046110, 63400366, 1935422, 49991118, 12418654, 5948750, 499806, 991582, 291758, 97374, 36910, 13166, 877959183, 527192095, 80842799, 57876543, 39352399, 5508191, 2612847, 2761855, 435599, 124575, 41647, 16319, 16335]); a,b,c,d,e,f,g,h,i,j,k,l := Explode([GPC.1, GPC.2, GPC.4, GPC.6, GPC.8, GPC.10, GPC.11, GPC.12, GPC.13, GPC.14, GPC.15, GPC.16]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "d", "d2", "e", "e2", "f", "g", "h", "i", "j", "k", "l"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 54911);

// 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