# Gap code for working with abstract group 1944.3566.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G := SmallGroup(1944, 3566);

# 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(13201371736240525713027266144715003929289493523803,1944); a := GPC.1; b := GPC.3; c := GPC.6; d := GPC.8;

# Define the group as a permutation group: 
Group( (2,4)(3,7)(5,6)(8,9), (10,11)(12,15)(13,14), (10,11)(12,14)(13,15), (2,5,9)(4,6,8), (10,12,13)(11,14,15), (10,13,12)(11,14,15), (1,2,6,7,9,8,3,5,4), (1,3,7)(2,5,9)(4,8,6) );

# Define the group as a matrix group with coefficients in GLZN: 
Group([[[ZmodnZObj(1,90), ZmodnZObj(0,90)], [ZmodnZObj(0,90), ZmodnZObj(61,90)]],[[ZmodnZObj(31,90), ZmodnZObj(80,90)], [ZmodnZObj(60,90), ZmodnZObj(61,90)]],[[ZmodnZObj(71,90), ZmodnZObj(80,90)], [ZmodnZObj(45,90), ZmodnZObj(1,90)]],[[ZmodnZObj(73,90), ZmodnZObj(54,90)], [ZmodnZObj(18,90), ZmodnZObj(1,90)]],[[ZmodnZObj(46,90), ZmodnZObj(45,90)], [ZmodnZObj(45,90), ZmodnZObj(1,90)]],[[ZmodnZObj(19,90), ZmodnZObj(54,90)], [ZmodnZObj(0,90), ZmodnZObj(1,90)]],[[ZmodnZObj(1,90), ZmodnZObj(30,90)], [ZmodnZObj(0,90), ZmodnZObj(1,90)]],[[ZmodnZObj(19,90), ZmodnZObj(0,90)], [ZmodnZObj(45,90), ZmodnZObj(19,90)]]]);

# Define the group from the transitive group database: 
TransitiveGroup(36, 2857);

# 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