// Magma code for working with abstract group 2916.md.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := PermutationGroup< 30 | (2,5)(3,9)(4,11,21,23,10,22)(6,12)(7,13)(8,17,16,25,15,26)(14,20,24,27,18,19)(28,29,30), (2,6,7)(4,8,18)(5,12,13)(10,15,24)(11,17,19)(14,21,16)(20,23,25)(22,26,27), (1,3,9)(2,6,7)(4,10,21)(5,13,12)(8,15,16)(11,23,22)(14,18,24)(17,25,26)(19,20,27), (2,7)(3,9)(4,11,21,23,10,22)(5,13)(8,19,16,20,15,27)(14,25,24,26,18,17)(28,30,29), (1,2,5)(3,6,13)(4,8,14)(7,12,9)(10,15,18)(11,20,17)(16,24,21)(19,26,22)(23,27,25), (4,10,21)(8,15,16)(11,22,23)(14,18,24)(17,26,25)(19,27,20)(28,29,30), (1,4,11,9,21,22,3,10,23)(2,8,20,7,16,19,6,15,27)(5,14,17,12,24,26,13,18,25)(28,30,29), (28,30,29) >;

// 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, -3, -2, -3, -3, -3, 3, -3, 16, 14546, 11746, 66, 72195, 51845, 17301, 8237, 36294, 36302, 7078, 2046, 222, 124423, 20759]); a,b,c,d,e := Explode([GPC.1, GPC.3, GPC.5, GPC.6, GPC.7]); AssignNames(~GPC, ["a", "a2", "b", "b2", "c", "d", "e", "e3"]);

// Define the group as a permutation group: 
PermutationGroup< 30 | (2,5)(3,9)(4,11,21,23,10,22)(6,12)(7,13)(8,17,16,25,15,26)(14,20,24,27,18,19)(28,29,30), (2,6,7)(4,8,18)(5,12,13)(10,15,24)(11,17,19)(14,21,16)(20,23,25)(22,26,27), (1,3,9)(2,6,7)(4,10,21)(5,13,12)(8,15,16)(11,23,22)(14,18,24)(17,25,26)(19,20,27), (2,7)(3,9)(4,11,21,23,10,22)(5,13)(8,19,16,20,15,27)(14,25,24,26,18,17)(28,30,29), (1,2,5)(3,6,13)(4,8,14)(7,12,9)(10,15,18)(11,20,17)(16,24,21)(19,26,22)(23,27,25), (4,10,21)(8,15,16)(11,22,23)(14,18,24)(17,26,25)(19,27,20)(28,29,30), (1,4,11,9,21,22,3,10,23)(2,8,20,7,16,19,6,15,27)(5,14,17,12,24,26,13,18,25)(28,30,29), (28,30,29) >;

// 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