# Gap code for working with abstract group 11664.kv.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(3736206736914572732371176538338573298327729471332021339675670395103035095875381889405422227048755497135150582706638795111865356981490165501023,11664); a := GPC.1; b := GPC.2; c := GPC.4; d := GPC.5; e := GPC.7; f := GPC.8; g := GPC.9; h := GPC.10;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 9560);

# 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