# Gap code for working with abstract group 432.520.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G := SmallGroup(432, 520);

# 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(315743925718442161789025476127419366510060295,432); a := GPC.1; b := GPC.2; c := GPC.5; d := GPC.6; e := GPC.7;

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

# Define the group as a matrix group with coefficients in Z: 
Group([[[-1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0], [-1, 1, 1, 0, 0, 0], [0, 0, -1, -1, -1, 0], [1, -1, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 1]], [[0, -1, 0, 0, 0, 0], [0, -1, 0, -1, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, -1, 0, -1, 1], [1, 0, 1, 0, 1, -1], [1, -1, 0, 0, 1, -1]]]);

# Define the group as a matrix group with coefficients in GLFp: 
Group([[[ Z(3)^0, 0*Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ Z(3), Z(3), Z(3), Z(3) ], [ Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0 ]], [[ Z(3)^0, Z(3)^0, 0*Z(3), Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3), 0*Z(3) ], [ 0*Z(3), Z(3), 0*Z(3), Z(3) ]], [[ Z(3), Z(3), Z(3), 0*Z(3) ], [ Z(3), Z(3)^0, 0*Z(3), Z(3) ], [ Z(3), Z(3), 0*Z(3), Z(3)^0 ], [ Z(3), Z(3)^0, Z(3)^0, Z(3)^0 ]], [[ Z(3), 0*Z(3), Z(3), 0*Z(3) ], [ Z(3), Z(3)^0, 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ Z(3), 0*Z(3), Z(3)^0, Z(3)^0 ]], [[ Z(3)^0, Z(3), 0*Z(3), 0*Z(3) ], [ Z(3), Z(3)^0, Z(3), Z(3)^0 ], [ 0*Z(3), Z(3)^0, Z(3), Z(3)^0 ], [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ]], [[ Z(3), 0*Z(3), Z(3), 0*Z(3) ], [ 0*Z(3), Z(3), 0*Z(3), 0*Z(3) ], [ Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0 ], [ Z(3), 0*Z(3), Z(3)^0, Z(3)^0 ]], [[ Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [ Z(3)^0, 0*Z(3), Z(3)^0, Z(3)^0 ], [ Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ]]]);

# Define the group from the transitive group database: 
TransitiveGroup(27, 141);
TransitiveGroup(36, 704);

# 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