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


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

// 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, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 26, 5532008, 130340, 8133504, 106, 5785731, 2791480, 1524786, 2394994, 4933907, 5049360, 392188, 186, 8592485, 620586, 307975, 103004, 304, 353814, 19675, 6584, 1647367, 20386100, 7772577, 3880702, 15035, 410, 25280, 227469, 75850, 12696, 33667929, 2541262, 847115, 65568, 63241, 91334, 56185282, 988439, 329508, 211117, 200834, 19380, 3639179, 8491416, 11574613, 707666, 454959, 61878, 22668996, 15313453, 5104514, 2223753, 949168, 442688]); a,b,c,d,e,f,g,h := Explode([GPC.1, GPC.3, GPC.5, GPC.8, GPC.10, GPC.11, GPC.12, GPC.13]); AssignNames(~GPC, ["a", "a2", "b", "b2", "c", "c2", "c6", "d", "d3", "e", "f", "g", "h"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 30262);

// 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