// Magma code for working with abstract group 311040.j.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := PermutationGroup< 81 | (1,56,49,12,42,35)(2,3,48)(4,18,78,21,81,36)(5,8,34,19,33,9)(6,25,44,38,39,69)(7,58,11)(10,14,67,53,20,75)(13,61,51,74,23,59)(15,27,32)(16,57,66,30,31,17)(22,55,65,72,70,40)(24,73,62,54,52,76)(26,60,29,77,47,41)(28,46,37,64,68,63)(43,71,45,80,50,79), (1,78,9,48,64,43,52,53)(2,3,4,10,19,81,62,49)(5,65,80,39,70,72,58,33)(6,30,37,28,77,71,40,60)(7,22,44,34,57,41,55,68)(8,46,75,14,45,67,24,50)(11,17)(13,36,21,15,35,31,27,74)(16,18,26,69,59,76,23,32)(25,56,61,38,66,47,73,54)(29,51)(42,63) >;

// 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 as a permutation group: 
PermutationGroup< 81 | (1,56,49,12,42,35)(2,3,48)(4,18,78,21,81,36)(5,8,34,19,33,9)(6,25,44,38,39,69)(7,58,11)(10,14,67,53,20,75)(13,61,51,74,23,59)(15,27,32)(16,57,66,30,31,17)(22,55,65,72,70,40)(24,73,62,54,52,76)(26,60,29,77,47,41)(28,46,37,64,68,63)(43,71,45,80,50,79), (1,78,9,48,64,43,52,53)(2,3,4,10,19,81,62,49)(5,65,80,39,70,72,58,33)(6,30,37,28,77,71,40,60)(7,22,44,34,57,41,55,68)(8,46,75,14,45,67,24,50)(11,17)(13,36,21,15,35,31,27,74)(16,18,26,69,59,76,23,32)(25,56,61,38,66,47,73,54)(29,51)(42,63) >;

// 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