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


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

// 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, 46190, 13901006, 183393, 106, 21902403, 3491712, 75858, 13871524, 11246837, 1613070, 758983, 251, 12282197, 2839, 723638, 8707627, 6674882, 3321, 634783, 34587911, 10378388, 6842193, 9406, 710483, 17245, 410, 1364696, 227469, 252729, 758182, 13353922, 25158299, 1251000, 221452, 73460, 36402923, 33813960, 4099717, 1026374, 417051, 266849, 88554, 12180, 13305746]); a,b,c,d,e,f,g,h,i := Explode([GPC.1, GPC.3, GPC.5, GPC.7, GPC.8, GPC.10, GPC.11, GPC.12, GPC.13]); AssignNames(~GPC, ["a", "a2", "b", "b2", "c", "c3", "d", "e", "e3", "f", "g", "h", "i"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 30298);

// 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