// Magma code for working with abstract group 314928.py.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2678832, 4928145, 66, 8188442, 12668763, 7953832, 2747345, 146, 13703044, 382997, 2760, 251, 202181, 5634, 5499318, 9621631, 63914, 1266219, 45103, 266, 2438599, 157268, 44961, 29998, 410, 16856, 25319, 18195849, 1010902, 884555, 18807, 516, 15011578, 15532, 4245707, 5458776, 5458789, 3639218, 202265, 622, 1971228, 17740969, 2956875, 164358]); a,b,c,d,e,f := Explode([GPC.1, GPC.2, GPC.4, GPC.7, GPC.10, GPC.12]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "c6", "d", "d2", "d6", "e", "e3", "f", "f3"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 27822);

// 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