# Gap code for working with abstract group 64.1.
# Some of these functions may take a long time to execute (this depends on the group).


# Define group as a cyclic group: 
G := CyclicGroup(64);

# 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(18159675375647,64); a := GPC.1;

# Define the group as a permutation group: 
Group( (1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33), (1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17)(33,64,48,56,40,60,44,52,36,62,46,54,38,58,42,50,34,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49), (1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)(17,32,24,28,20,30,22,26,18,31,23,27,19,29,21,25)(33,48,40,44,36,46,38,42,34,47,39,43,35,45,37,41)(49,64,56,60,52,62,54,58,50,63,55,59,51,61,53,57), (1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)(17,24,20,22,18,23,19,21)(25,32,28,30,26,31,27,29)(33,40,36,38,34,39,35,37)(41,48,44,46,42,47,43,45)(49,56,52,54,50,55,51,53)(57,64,60,62,58,63,59,61), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64) );

# Define the group as a matrix group with coefficients in GLFp: 
Group([[[ Z(31)^27, Z(31) ], [ Z(31)^0, Z(31)^27 ]]]);

# 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