// Magma code for working with abstract group 384.16394.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := SmallGroup(384, 16394);

// 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([8, -2, -2, -2, -2, -2, -2, -2, -3, 730, 66, 907, 91, 972, 116, 4502, 166, 4119]); a,b,c,d := Explode([GPC.1, GPC.2, GPC.3, GPC.7]); AssignNames(~GPC, ["a", "b", "c", "c2", "c4", "c8", "d", "d2"]);

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

// Define the group as a matrix group with coefficients in GLZN: 
MatrixGroup< 2, Integers(48) | [[1, 6, 0, 1], [31, 0, 0, 1], [7, 47, 0, 41], [1, 24, 0, 1], [47, 0, 0, 47], [1, 16, 0, 1], [17, 0, 0, 17], [1, 12, 0, 1]] >;

// 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