# Gap code for working with abstract group 30233088.jb.
# Some of these functions may take a long time to execute (this depends on the group).


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

# Order of the group: 
Order(G);

# Exponent of the group: 
Exponent(G);

# Automorphism group: 
AutomorphismGroup(G);

# The outer automorphism group of G: 
FactorGroup(AutomorphismGroup(G), InnerAutomorphismGroup(G));

# Composition factors of the group: 
CompositionSeries(G);

# Nilpotency class of the group: 
if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;

# 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 a monomial group: 
IsMonomialGroup(G);

# Determine if the group G is nilpotent: 
IsNilpotentGroup(G);

# Determine if the group G is perfect: 
IsPerfectGroup(G);

# Determine if the group G is a p-group: 
IsPGroup(G);

# Determine if the group G is polycyclic: 
IsPolycyclicGroup(G);

# Determine if the group G is simple: 
IsSimpleGroup(G);

# Determine if the group G is solvable: 
IsSolvableGroup(G);

# Determine if the group G is supersolvable: 
IsSupersolvableGroup(G);

# Compute statistics for the group G: 
# Gap code to output the first two rows of the group statistics table
element_orders := List(Elements(G), g -> Order(g));
orders := Set(element_orders);
Print("Orders: ", orders, "\n");
element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n)));
Print("Elements: ", element_counts, " ", Size(G), "\n");
cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc)));
cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n)));
Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");

# 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 := PcGroupCode(1010661459502073343500333155487583221862808541257019037188358118431622438246773493609899205026339510409025894802333024526210336513878779168695261791793578332386721935298485335721383177157908846571017012623976044427746225016252031912799185518189004381952322969220690634965047595739056633592057990214092222228964663997048341807974893873418184578553152239650507548963500511477514909491301588598303266294094541102439079800770438323749001832089907856606820953482663489949051591510639079961937753624764273884660079990093081940021007077954965767551982991501351160342815390501663026759124449168465733777606896147766550240256698900383839960617581347966188535772218667531259725451305707986587042111435966803734202222291138258797272114079152845181606194808252057984987188860737426981418871344496161874068291865588232455131535796802523202328329761621905023339355738486922489688546343961265320259205961138021536898852595260118847605389660388336756081097808969846096130090298415362752056294902363997127380768738088806907074264434090399707041618633812931551871,30233088); a := GPC.1; b := GPC.3; c := GPC.5; d := GPC.7; e := GPC.9; f := GPC.11; g := GPC.14; h := GPC.15; i := GPC.16; j := GPC.17; k := GPC.18; l := GPC.19;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 70115);

# The primary decomposition of the group: 
AbelianInvariants(G);

# The abelianization of the group: 
FactorGroup(G, DerivedSubgroup(G));

# The Schur multiplier of the group: 
AbelianInvariantsMultiplier(G);

# The commutator length of the group: 
CommutatorLength(G);

# List of subgroups of the group: 
AllSubgroups(G);

# Center of the group: 
Center(G);

# Commutator subgroup of the group G: 
DerivedSubgroup(G);

# Frattini subgroup of the group G: 
FrattiniSubgroup(G);

# Fitting subgroup of the group G: 
FittingSubgroup(G);

# Radical of the group G: 
SolvableRadical(G);

# Socle of the group G: 
Socle(G);

# Derived series of the group G: 
DerivedSeriesOfGroup(G);

# Chief series of the group G: 
ChiefSeries(G);

# The lower central series of the group G: 
LowerCentralSeriesOfGroup(G);

# The upper central series of the group G: 
UpperCentralSeriesOfGroup(G);

# Character table: 
CharacterTable(G);  # Output not guaranteed to exactly match the LMFDB table