// Magma code for working with abstract group 1417176.fq.
// Some of these functions may take a long time to execute (this depends on the group).


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

// 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([14, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 28, 19194590, 16521766, 114, 62864259, 11757665, 20293564, 48468438, 1284812, 8136076, 200, 65288165, 53691139, 19391937, 36218454, 3154052, 21951250, 1559438, 153885319, 78357909, 19164131, 3580465, 3169215, 796061, 33691, 54537, 40170824, 19962958, 4735620, 1503748, 83266, 30209769, 14696663, 635105, 35387, 155509210, 6586296, 31356902, 1970958, 91584, 28413515, 23024761, 10287687, 54541, 57316908, 9552856, 1592204, 203805517, 13420539, 22889705, 10351207, 7166613, 3285827, 486961, 225903, 70097, 26599, 10149, 3107]); a,b,c,d,e,f,g,h,i,j,k := Explode([GPC.1, GPC.3, GPC.5, GPC.7, GPC.8, GPC.9, GPC.10, GPC.11, GPC.12, GPC.13, GPC.14]); AssignNames(~GPC, ["a", "a2", "b", "b2", "c", "c2", "d", "e", "f", "g", "h", "i", "j", "k"]);

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

// Define the group from the transitive group database: 
TransitiveGroup(36, 40159);

// 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