// Magma code for working with abstract group 11664.kv.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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, 3, 2, 3, 3, 3, 3, 3, 112800, 122201, 51, 260762, 229123, 62733, 103863, 562804, 137614, 1524, 41584, 144, 158405, 49455, 1465, 39275, 624966, 403216, 186506, 27346, 2156, 864007, 28817, 120987, 56197, 20207, 2937, 4627, 116658, 9768, 777609]); a,b,c,d,e,f,g,h := Explode([GPC.1, GPC.2, GPC.4, GPC.5, GPC.7, GPC.8, GPC.9, GPC.10]); AssignNames(~GPC, ["a", "b", "b2", "c", "d", "d2", "e", "f", "g", "h"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 9560);

// 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