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


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

// 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, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 32, 76901393, 43586, 11814450, 16020898, 185147395, 139739411, 57723555, 179, 408770564, 97079700, 110897956, 8708, 98936069, 225513237, 28881829, 4789493, 18203109, 277, 334656006, 241731862, 749990, 2288886, 1149862, 489990, 200073223, 130019351, 49545255, 85879351, 3368519, 1001559, 500455, 503, 60466184, 839848, 5272, 546462729, 235630105, 1904681, 121017, 18921673, 8568089, 27481, 116793, 600652810, 171756314, 120143274, 56111674, 598826, 228186, 1159610, 300058, 698, 725594123, 4479003, 41515, 186715, 863875596, 31629340, 66763052, 15769788, 6368620, 610364, 153612, 202300, 387085, 211631661, 11757373, 604661774, 305130270, 9901486, 37791422, 15746494, 62798, 77934, 5971983, 483729439, 26873903, 120932415]); a,b,c,d,e,f,g,h,i,j,k := Explode([GPC.1, GPC.3, GPC.4, GPC.6, GPC.8, GPC.10, GPC.11, GPC.13, GPC.14, GPC.15, GPC.16]); AssignNames(~GPC, ["a", "a2", "b", "c", "c2", "d", "d2", "e", "e3", "f", "g", "g3", "h", "i", "j", "k"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 54993);

// 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