# Gap code for working with abstract group 64.54.
# Some of these functions may take a long time to execute (this depends on the group).


# Define group as a dicyclic group: 
G := DicyclicGroup(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 group as Lie representations: 
Group([[[ Z(31)^29, Z(31)^8 ], [ Z(31)^24, Z(31)^15 ]], [[ Z(31)^15, Z(31)^23 ], [ Z(31)^9, Z(31)^29 ]], [[ Z(31)^15, 0*Z(31) ], [ 0*Z(31), Z(31)^15 ]], [[ Z(31)^26, Z(31)^10 ], [ Z(31)^26, Z(31)^11 ]]]);

# Define the group with the given generators and relations: 
GPC := PcGroupCode(309504273051162771967,64); a := GPC.1; b := GPC.2;

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