# Gap code for working with abstract group 30576.b.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G := Group( (1,2,3,6)(4,7,9,14)(5,12,10,20)(8,17,15,28)(11,22,19,24)(13,25,21,34)(16,30,27,41)(18,32,29,43)(23,36,33,45)(26,39,35,48)(31,44,42,54)(37,51,46,56)(38,50,47,55)(40,53,49,52)(57,58,61,60,62,63,59), (1,3)(2,6)(4,9)(5,10)(7,14)(8,15)(11,19)(12,20)(13,21)(16,27)(17,28)(18,29)(22,24)(23,33)(25,34)(26,35)(30,41)(31,42)(32,43)(36,45)(37,46)(38,47)(39,48)(40,49)(44,54)(50,55)(51,56)(52,53)(57,59,63,62,60,61,58), (1,4,3,9)(2,7,6,14)(5,11,10,19)(8,16,15,27)(12,22,20,24)(13,23,21,33)(17,30,28,41)(18,31,29,42)(25,36,34,45)(26,37,35,46)(32,44,43,54)(38,50,47,55)(39,51,48,56)(40,52,49,53)(57,60,59,61,63,58,62), (57,58,61,60,62,63,59), (1,5,13,26,40,6,15,29,28,43,36,50,51,9,19,33,46,53,7,16,31,30,44,25,38,39,3,10,21,35,49,2,8,18,17,32,45,55,56,4,11,23,37,52,14,27,42,41,54,34,47,48)(12,24,20,22)(57,61,62,59,58,60,63) );

# 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( (1,2,3,6)(4,7,9,14)(5,12,10,20)(8,17,15,28)(11,22,19,24)(13,25,21,34)(16,30,27,41)(18,32,29,43)(23,36,33,45)(26,39,35,48)(31,44,42,54)(37,51,46,56)(38,50,47,55)(40,53,49,52)(57,58,61,60,62,63,59), (1,3)(2,6)(4,9)(5,10)(7,14)(8,15)(11,19)(12,20)(13,21)(16,27)(17,28)(18,29)(22,24)(23,33)(25,34)(26,35)(30,41)(31,42)(32,43)(36,45)(37,46)(38,47)(39,48)(40,49)(44,54)(50,55)(51,56)(52,53)(57,59,63,62,60,61,58), (1,4,3,9)(2,7,6,14)(5,11,10,19)(8,16,15,27)(12,22,20,24)(13,23,21,33)(17,30,28,41)(18,31,29,42)(25,36,34,45)(26,37,35,46)(32,44,43,54)(38,50,47,55)(39,51,48,56)(40,52,49,53)(57,60,59,61,63,58,62), (57,58,61,60,62,63,59), (1,5,13,26,40,6,15,29,28,43,36,50,51,9,19,33,46,53,7,16,31,30,44,25,38,39,3,10,21,35,49,2,8,18,17,32,45,55,56,4,11,23,37,52,14,27,42,41,54,34,47,48)(12,24,20,22)(57,61,62,59,58,60,63) );

# Define the group as a matrix group with coefficients in GLFq: 
Group([[[Z(169)^25, Z(169)^126], [Z(169)^164, Z(169)^109]],[[Z(169)^132, 0*Z(169)], [0*Z(169), Z(169)^132]],[[Z(169)^150, 0*Z(169)], [0*Z(169), Z(169)^150]],[[Z(169)^120, 0*Z(169)], [0*Z(169), Z(169)^120]],[[Z(169)^137, Z(169)^87], [Z(169)^51, Z(169)^75]]]);

# 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