// Magma code for working with abstract group 104976.pc.
// Some of these functions may take a long time to execute (this depends on the group).


// Construction of abstract group: 
G := PermutationGroup< 36 | (1,28,34,26,5,12)(2,30,36,25,6,10)(3,29,35,27,4,11)(7,31,8,32,9,33)(13,16,22)(14,17,23,15,18,24)(20,21), (1,20,12,18,13,9,35,29)(2,19,11,16,14,7,36,28)(3,21,10,17,15,8,34,30)(4,27,31,24)(5,26,33,22,6,25,32,23) >;

// 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([12, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 24, 1884170, 158234, 303722, 481251, 1787199, 668619, 116199, 4154764, 2221576, 818848, 235120, 172, 3582149, 158129, 956477, 312521, 96774, 1102782, 2730, 174102, 402, 1246495, 5419, 22327, 1219, 10793096, 489908, 979808, 207404, 291656, 15620, 428, 9331209, 699873, 518445, 233337, 13029, 6291658, 5388790, 510082, 170062, 261418, 171142, 526, 1026467, 342191, 15611, 140039]); a,b,c,d,e,f,g,h := Explode([GPC.1, GPC.3, GPC.4, GPC.5, GPC.7, GPC.8, GPC.9, GPC.11]); AssignNames(~GPC, ["a", "a2", "b", "c", "d", "d2", "e", "f", "g", "g3", "h", "h3"]);

// Define the group as a permutation group: 
PermutationGroup< 36 | (1,28,34,26,5,12)(2,30,36,25,6,10)(3,29,35,27,4,11)(7,31,8,32,9,33)(13,16,22)(14,17,23,15,18,24)(20,21), (1,20,12,18,13,9,35,29)(2,19,11,16,14,7,36,28)(3,21,10,17,15,8,34,30)(4,27,31,24)(5,26,33,22,6,25,32,23) >;

// Define the group from the transitive group database: 
TransitiveGroup(36, 20326);

// 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