// Magma code for working with abstract group 229582512.bu.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([19, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 38, 2396434718, 498584360, 4091918529, 1265227672, 16506268851, 1785724978, 390603029, 268420812, 18673577914, 1795703513, 339741987, 209547831, 270, 21353809685, 8047192992, 3121678915, 152966630, 328, 12520581702, 5208032113, 2396125916, 1094452807, 21436012807, 15919448858, 3772105005, 148933312, 1333427, 440434086, 19956840920, 17846441019, 3631198726, 177535001, 511032, 511051, 378963369, 3975791068, 16621247, 117456546, 1108165, 20624, 12760914682, 16780656845, 5177103936, 2383156843, 155942510, 53522079, 26793088, 8539883, 341250, 106144, 827, 21325664459, 17663681694, 361677361, 981315716, 443319, 49775175228, 2281758367, 160106, 1494763053, 26764, 720359, 22432, 53606103277, 2568972704, 4826355, 1609802278, 1536861905, 2327076, 201299, 46313311694, 1480948953, 46539412, 2572961831, 7756650, 11494877199, 11655425698, 7642370357, 1431847368, 1507098331, 613728686, 121297953, 106104964, 714263, 520045, 13923, 12118, 52804887568, 103605534, 2933604937, 1466802524, 70601982065, 3127001796, 9022061287, 554384810, 64317981, 955054264, 230702387, 130587378, 6787159, 1277577, 89507, 38226, 14666, 31049731314, 3252768877, 5745505664, 1914466827, 1544977570]); a,b,c,d,e,f,g,h,i,j,k,l,m,n,o := Explode([GPC.1, GPC.3, GPC.4, GPC.5, GPC.8, GPC.9, GPC.10, GPC.11, GPC.13, GPC.14, GPC.15, GPC.16, GPC.17, GPC.18, GPC.19]); AssignNames(~GPC, ["a", "a2", "b", "c", "d", "d2", "d4", "e", "f", "g", "h", "h3", "i", "j", "k", "l", "m", "n", "o"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 83687);

// 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