// Magma code for working with abstract group 1296.2891.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := SmallGroup(1296, 2891);

// 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, 2, -2, -3, 3, -3, 65, 1010, 226, 114, 1155, 107, 211, 91, 23045, 8469, 2333, 1957, 16134, 17486, 3158, 2494, 710, 718, 55303]); a,b,c,d,e,f,g := Explode([GPC.1, GPC.2, GPC.3, GPC.4, GPC.6, GPC.7, GPC.8]); AssignNames(~GPC, ["a", "b", "c", "d", "d2", "e", "f", "g"]);

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

// Define the group as a matrix group with coefficients in Z: 
MatrixGroup< 6, Integers() | [[0, 1, 1, 0, 0, -1, 1, 0, 1, 0, 1, -1, -1, 1, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0], [1, 0, 0, 0, 1, -1, 1, 0, 0, 1, 1, 0, 1, -1, 0, -1, 0, -1, -1, 0, -1, -1, -1, 1, 0, 0, 1, 1, 1, 0, 1, -1, 0, 0, 1, -1]] >;

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

// Define the group from the transitive group database: 
TransitiveGroup(27, 294);
TransitiveGroup(36, 2302);

// 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