// Magma code for working with abstract group 15552.lb.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([11, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 4752, 62261, 56, 414878, 626739, 106406, 101665, 7956, 574204, 406905, 169481, 1247, 158, 700133, 58624, 99027, 29870, 1057062, 456011, 286699, 98291, 13910, 23007, 226, 946183, 177426, 270365, 105640, 38067, 766, 1368584, 47539, 52302, 42809, 59452, 2439, 1262, 1224969, 517460, 327391, 54602, 70453, 7984, 9975, 673738, 46485, 151040, 11659, 1022, 26201]); a,b,c,d,e,f,g,h := Explode([GPC.1, GPC.2, GPC.4, GPC.5, GPC.7, GPC.9, GPC.10, GPC.11]); AssignNames(~GPC, ["a", "b", "b2", "c", "d", "d2", "e", "e2", "f", "g", "h"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 10243);
TransitiveGroup(36, 10251);

// 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