// Magma code for working with abstract group 6552.b.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := PermutationGroup< 59 | (1,2,5)(3,8,9)(4,11,23)(10,20,36)(12,25,42)(13,18,32)(14,28,30)(15,17,26)(16,24,41)(19,33,22)(21,38,37)(27,39,44)(29,45,54)(31,46,43)(40,52,47)(48,53,49)(57,58,59), (1,3,9,18,32,26,5,13,2,6,15,8,17)(4,12,25,20,37,42,38,50,36,23,10,21,11)(14,29,33,28,19,34,45,52,54,22,40,47,30)(16,27,24,31,44,39,49,48,41,43,53,46,55), (1,4,2,7,16,30,41,5,14,28,23,11,24)(3,10,22,9,19,35,36,49,20,33,48,53,8)(12,26,43,25,42,52,40,17,31,47,46,56,15)(13,27,44,54,21,39,38,51,45,18,29,37,32) >;

// 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 as a permutation group: 
PermutationGroup< 59 | (1,2,5)(3,8,9)(4,11,23)(10,20,36)(12,25,42)(13,18,32)(14,28,30)(15,17,26)(16,24,41)(19,33,22)(21,38,37)(27,39,44)(29,45,54)(31,46,43)(40,52,47)(48,53,49)(57,58,59), (1,3,9,18,32,26,5,13,2,6,15,8,17)(4,12,25,20,37,42,38,50,36,23,10,21,11)(14,29,33,28,19,34,45,52,54,22,40,47,30)(16,27,24,31,44,39,49,48,41,43,53,46,55), (1,4,2,7,16,30,41,5,14,28,23,11,24)(3,10,22,9,19,35,36,49,20,33,48,53,8)(12,26,43,25,42,52,40,17,31,47,46,56,15)(13,27,44,54,21,39,38,51,45,18,29,37,32) >;

// Define the group as a matrix group with coefficients in GLFp: 
MatrixGroup< 2, GF(13) | [[1, 1, 0, 1], [1, 0, 0, 3], [1, 0, 1, 1]] >;

// 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