// Magma code for working with abstract group 1296.855.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := SmallGroup(1296, 855);

// 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, 3, 3, 3, 2, 2, 3, 161, 41, 194, 7883, 3571, 123, 1452, 58757, 29389, 1749, 141, 64518, 32270, 4054, 166, 57607, 28815, 7511]); a,b,c,d := Explode([GPC.1, GPC.2, GPC.4, GPC.6]); AssignNames(~GPC, ["a", "b", "b2", "c", "c3", "d", "d2", "d4"]);

// Define the group as a permutation group: 
PermutationGroup< 31 | (2,5)(3,10)(4,13)(6,9)(7,14)(8,20)(11,23)(12,24)(15,22)(16,19)(17,25)(18,21)(26,27)(28,29), (3,11)(7,17)(8,19)(10,23)(12,24)(14,25)(16,20)(18,26)(21,27)(30,31), (28,30,29,31), (28,29)(30,31), (1,2,6,13,22,15,4,9,5)(3,8,16,23,17,26,12,21,14)(7,18,24,27,25,11,19,20,10), (2,7,17)(3,12,23)(5,14,25)(6,16,20)(8,9,19)(10,24,11)(15,26,18)(21,22,27), (1,3,10)(2,8,7)(4,12,11)(5,14,20)(6,16,18)(9,21,19)(13,23,24)(15,26,25)(17,27,22), (1,4,13)(2,9,22)(3,12,23)(5,15,6)(7,19,27)(8,21,17)(10,11,24)(14,26,16)(18,20,25) >;

// 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