// Magma code for working with abstract group 3888.el.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([9, -2, -3, -2, -2, -2, -3, 3, -3, 3, 18, 69176, 31196, 74, 34131, 58980, 102, 128524, 51313, 25925, 24638, 34151, 17312, 257, 102822, 58983, 10104, 5829, 798, 25945, 7810, 2635, 209960, 46682, 5867, 8792]); a,b,c,d,e,f := Explode([GPC.1, GPC.3, GPC.6, GPC.7, GPC.8, GPC.9]); AssignNames(~GPC, ["a", "a2", "b", "b2", "b4", "c", "d", "e", "f"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(27, 542);
TransitiveGroup(36, 4854);
TransitiveGroup(36, 4958);

// 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