// Magma code for working with abstract group 216.155.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := SmallGroup(216, 155);

// 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([6, -2, -2, -2, -3, -3, 3, 12, 31, 387, 2884, 1810, 376, 869, 1739, 1313]); a,b,c,d := Explode([GPC.1, GPC.4, GPC.5, GPC.6]); AssignNames(~GPC, ["a", "a2", "a4", "b", "c", "d"]);

// Define the group as a permutation group: 
PermutationGroup< 12 | (2,4,5,7,6,8,9,3)(11,12), (2,5,6,9)(3,4,7,8), (2,6)(3,7)(4,8)(5,9), (1,2,6)(3,4,5)(7,9,8), (1,3,7)(2,4,9)(5,8,6), (10,11,12) >;

// Define the group as a matrix group with coefficients in GLFp: 
MatrixGroup< 4, GF(3) | [[2, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0], [1, 1, 2, 2, 0, 2, 0, 0, 0, 1, 1, 0, 0, 2, 1, 2], [2, 1, 1, 2, 2, 1, 0, 2, 1, 1, 2, 2, 2, 2, 2, 2], [0, 2, 2, 1, 2, 0, 1, 0, 2, 0, 1, 2, 2, 1, 1, 1], [0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 1, 2, 1, 2, 1, 0], [1, 0, 2, 2, 2, 1, 0, 2, 1, 0, 0, 0, 0, 0, 1, 2]] >;

// Define the group from the transitive group database: 
TransitiveGroup(24, 568);
TransitiveGroup(27, 80);
TransitiveGroup(36, 288);

// 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