// Magma code for working with abstract group 472392.ta.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([13, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 26, 5040518, 8688825, 106, 18208739, 6509376, 119382, 27950524, 8046497, 3942540, 1830703, 35901221, 8353818, 440887, 1462557, 304, 3538086, 19675, 9860, 3863815, 67412, 1033377, 4675328, 227469, 69532, 29820969, 631822, 8094095, 400188, 221191, 74954, 3027034, 2501951, 1328220, 24209963, 16308888, 3691621, 1276286, 369315, 212080, 2000, 12300508, 8140417, 5932952, 4271019, 1385695, 480713, 5706]); a,b,c,d,e,f,g,h,i,j := Explode([GPC.1, GPC.3, GPC.5, GPC.6, GPC.8, GPC.9, GPC.10, GPC.11, GPC.12, GPC.13]); AssignNames(~GPC, ["a", "a2", "b", "b2", "c", "d", "d3", "e", "f", "g", "h", "i", "j"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 30313);

// 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