# Gap code for working with abstract group 6552.b.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G := Group( (1,2,5)(3,8,9)(4,11,23)(10,20,36)(12,25,42)(13,18,32)(14,28,30)(15,17,26)(16,24,41)(19,33,22)(21,38,37)(27,39,44)(29,45,54)(31,46,43)(40,52,47)(48,53,49)(57,58,59), (1,3,9,18,32,26,5,13,2,6,15,8,17)(4,12,25,20,37,42,38,50,36,23,10,21,11)(14,29,33,28,19,34,45,52,54,22,40,47,30)(16,27,24,31,44,39,49,48,41,43,53,46,55), (1,4,2,7,16,30,41,5,14,28,23,11,24)(3,10,22,9,19,35,36,49,20,33,48,53,8)(12,26,43,25,42,52,40,17,31,47,46,56,15)(13,27,44,54,21,39,38,51,45,18,29,37,32) );

# 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 as a permutation group: 
Group( (1,2,5)(3,8,9)(4,11,23)(10,20,36)(12,25,42)(13,18,32)(14,28,30)(15,17,26)(16,24,41)(19,33,22)(21,38,37)(27,39,44)(29,45,54)(31,46,43)(40,52,47)(48,53,49)(57,58,59), (1,3,9,18,32,26,5,13,2,6,15,8,17)(4,12,25,20,37,42,38,50,36,23,10,21,11)(14,29,33,28,19,34,45,52,54,22,40,47,30)(16,27,24,31,44,39,49,48,41,43,53,46,55), (1,4,2,7,16,30,41,5,14,28,23,11,24)(3,10,22,9,19,35,36,49,20,33,48,53,8)(12,26,43,25,42,52,40,17,31,47,46,56,15)(13,27,44,54,21,39,38,51,45,18,29,37,32) );

# Define the group as a matrix group with coefficients in GLFp: 
Group([[[ Z(13)^0, Z(13)^0 ], [ 0*Z(13), Z(13)^0 ]], [[ Z(13)^0, 0*Z(13) ], [ 0*Z(13), Z(13)^4 ]], [[ Z(13)^0, 0*Z(13) ], [ Z(13)^0, Z(13)^0 ]]]);

# 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