// Magma code for working with abstract group 972.115.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := SmallGroup(972, 115);

// 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([7, -2, -2, -3, -3, -3, 3, -3, 141, 36, 170, 12946, 6401, 108, 12611, 6003, 36293, 12108, 4933, 166, 31758, 15889]); a,b,c,d := Explode([GPC.1, GPC.2, GPC.4, GPC.6]); AssignNames(~GPC, ["a", "b", "b2", "c", "c3", "d", "d3"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(27, 287);

// 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