// Magma code for working with abstract group 500.5.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := SmallGroup(500, 5);

// 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([5, -2, -2, -5, -5, -5, 26, 87, 118]); a,b := Explode([GPC.1, GPC.2]); AssignNames(~GPC, ["a", "b", "b2", "b10", "b50"]);

// Define the group as a permutation group: 
PermutationGroup< 129 | (1,2), (3,4), (5,129,104,79,54,29,124,99,74,49,24,119,94,69,44,19,114,89,64,39,14,109,84,59,34,9,128,103,78,53,28,123,98,73,48,23,118,93,68,43,18,113,88,63,38,13,108,83,58,33,8,127,102,77,52,27,122,97,72,47,22,117,92,67,42,17,112,87,62,37,12,107,82,57,32,7,126,101,76,51,26,121,96,71,46,21,116,91,66,41,16,111,86,61,36,11,106,81,56,31,6,125,100,75,50,25,120,95,70,45,20,115,90,65,40,15,110,85,60,35,10,105,80,55,30), (5,29,24,19,14,9,28,23,18,13,8,27,22,17,12,7,26,21,16,11,6,25,20,15,10)(30,54,49,44,39,34,53,48,43,38,33,52,47,42,37,32,51,46,41,36,31,50,45,40,35)(55,79,74,69,64,59,78,73,68,63,58,77,72,67,62,57,76,71,66,61,56,75,70,65,60)(80,104,99,94,89,84,103,98,93,88,83,102,97,92,87,82,101,96,91,86,81,100,95,90,85)(105,129,124,119,114,109,128,123,118,113,108,127,122,117,112,107,126,121,116,111,106,125,120,115,110), (5,9,8,7,6)(10,14,13,12,11)(15,19,18,17,16)(20,24,23,22,21)(25,29,28,27,26)(30,34,33,32,31)(35,39,38,37,36)(40,44,43,42,41)(45,49,48,47,46)(50,54,53,52,51)(55,59,58,57,56)(60,64,63,62,61)(65,69,68,67,66)(70,74,73,72,71)(75,79,78,77,76)(80,84,83,82,81)(85,89,88,87,86)(90,94,93,92,91)(95,99,98,97,96)(100,104,103,102,101)(105,109,108,107,106)(110,114,113,112,111)(115,119,118,117,116)(120,124,123,122,121)(125,129,128,127,126) >;

// Define the group as a matrix group with coefficients in GLFp: 
MatrixGroup< 2, GF(251) | [[137, 0, 0, 66], [250, 0, 0, 250]] >;

// 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