// Magma code for working with abstract group 1285608.a.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := PSL(2,137);

// 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 group as Lie representations: 
PSL(2,137);
PSU(2,137);
Omega(3,137);
POmega(3,137);
PSigmaL(2,137);

// Define the group as a permutation group: 
PermutationGroup< 138 | (1,71,2)(3,86,124)(4,7,10)(5,132,24)(6,41,116)(8,36,111)(9,128,20)(11,28,66)(12,81,13)(14,88,84)(15,49,50)(16,125,104)(17,56,129)(18,109,70)(19,133,76)(21,101,126)(22,127,60)(23,96,135)(25,130,92)(26,51,131)(27,136,48)(29,69,100)(30,35,113)(31,89,105)(32,114,78)(33,55,42)(34,61,65)(37,45,98)(38,120,74)(39,117,122)(40,77,57)(43,134,82)(44,85,94)(46,90,80)(47,63,121)(52,83,123)(53,59,79)(54,107,115)(58,67,108)(62,106,72)(64,138,68)(73,93,99)(75,112,95)(87,91,118)(97,119,110)(102,103,137), (3,123,111,7,76,71,85,95,92,77,124,28,14,102,22,16,118,59,66,94,11,34,51,83,44,52,54,117,116,29,50,121,96,41,109,79,72,68,128,86,37,35,81,103,129,26,67,132,43,120,131,105,78,40,19,57,6,80,27,133,136,137,126,33,99,53,110,48)(4,5,8,114,61,135,84,122,101,63,36,10,21,98,9,74,115,12,38,60,106,104,55,13,73,69,62,32,100,45,20,91,112,25,24,87,89,97,58,90,107,130,47,75,82,23,125,119,39,127,113,17,64,49,46,56,70,65,134,30,18,138,93,31,88,42,108,15) >;

// 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