// Magma code for working with abstract group 7776.cw.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([10, -2, -2, -2, -2, -2, -3, 3, -3, -3, 3, 60921, 51, 59882, 82, 38723, 170014, 77424, 5934, 144, 85455, 28825, 515, 73926, 159056, 76186, 1716, 8026, 184327, 23057, 24527, 11577, 1747, 116658, 136108, 19478, 9768, 691209, 302419, 172829, 64839, 32449]); a,b,c,d,e,f,g := Explode([GPC.1, GPC.2, GPC.5, GPC.7, GPC.8, GPC.9, GPC.10]); AssignNames(~GPC, ["a", "b", "b2", "b4", "c", "c2", "d", "e", "f", "g"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(27, 723);
TransitiveGroup(36, 7343);
TransitiveGroup(36, 7367);

// 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