// Magma code for working with abstract group 3072.bju.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([11, -2, -2, -3, -2, -2, -2, -2, 2, 2, 2, 2, 1056, 2333, 56, 266, 806, 124, 3315, 268, 4768, 192, 2152, 4001, 22307]); a,b,c,d,e,f,g,h := Explode([GPC.1, GPC.2, GPC.4, GPC.6, GPC.8, GPC.9, GPC.10, GPC.11]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "d", "d2", "e", "f", "g", "h"]);

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

// 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