// Magma code for working with abstract group 384.5269.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := SmallGroup(384, 5269);

// 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, -3, -2, -2, -2, 16, 1545, 674, 2362, 6403, 91, 4053, 141, 4054, 166]); a,b,c,d := Explode([GPC.1, GPC.3, GPC.4, GPC.6]); AssignNames(~GPC, ["a", "a2", "b", "c", "c2", "d", "d2", "d4"]);

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

// 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