# Gap code for working with abstract group 1944.3475.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G := SmallGroup(1944, 3475);

# 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(376789862987692141402359691416409524917849242751546891930829599,1944); a := GPC.1; b := GPC.4; c := GPC.5; d := GPC.6; e := GPC.7; f := GPC.8;

# Define the group as a permutation group: 
Group( (2,5,6,16,7,17,20,8)(3,10,11,23,12,22,18,9)(4,15)(13,25,19,26,21,27,14,24)(29,30,31,32,33,35,36,34), (2,6,7,20)(3,11,12,18)(5,16,17,8)(9,10,23,22)(13,19,21,14)(24,25,26,27)(29,31,33,36)(30,32,35,34), (2,7)(3,12)(5,17)(6,20)(8,16)(9,23)(10,22)(11,18)(13,21)(14,19)(24,26)(25,27)(29,33)(30,35)(31,36)(32,34), (28,29,33)(30,34,36)(31,32,35), (28,30,35)(29,34,31)(32,33,36), (1,2,7)(3,13,20)(4,9,23)(5,18,22)(6,21,12)(8,24,19)(10,11,17)(14,26,16)(15,25,27)(28,31,36)(29,32,30)(33,35,34), (1,3,14)(2,8,6)(4,5,12)(7,22,11)(9,13,10)(15,19,17)(16,23,24)(18,26,25)(20,21,27)(28,32,34)(29,35,36)(30,33,31), (1,4,15)(2,9,25)(3,5,19)(6,10,26)(7,23,27)(8,13,18)(11,16,21)(12,17,14)(20,22,24) );

# 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