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


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

// 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, 55549440, 36406977, 81, 153322370, 46576515, 43473811, 23201507, 179, 69124, 7700, 3396, 471203141, 202151829, 54122725, 11225717, 277, 149313030, 16150, 595456519, 92924951, 44167719, 1764919, 169815, 375, 3464, 55862808, 27900328, 20792, 3544, 530703369, 15344665, 1969961, 69177, 11609, 2041, 1529098, 326633498, 116062890, 861754, 38106, 23354, 635682827, 17915931, 7838251, 746555, 124507, 20859, 1147461132, 14556700, 2426172, 404444, 67516, 34513933, 440100893, 169779501, 19192381, 1306461, 134525, 288720014, 14826270, 87194926, 25194302, 4199134, 390366, 401092623, 96989215, 155437103, 56365119, 13437023, 1230463]); 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 | (1,8,15,21,25,32)(2,9,14,19,26,31)(3,7,13,20,27,33)(4,29,16,5,30,17)(6,28,18)(10,36,24)(11,34,23,12,35,22), (1,18,19,36,3,17,21,35,2,16,20,34)(4,31,24,13,6,32,23,14,5,33,22,15)(7,11,27,29)(8,10,25,28)(9,12,26,30), (1,35,2,34,3,36)(4,32)(5,33)(6,31)(7,30)(8,28)(9,29)(10,25,11,26,12,27)(13,22,15,24,14,23)(16,20)(17,19)(18,21) >;

// Define the group from the transitive group database: 
TransitiveGroup(36, 54681);

// 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