# Gap code for working with abstract group 49800.b.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G := Group( (1,2,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82,81,83,78,80,75,77,72,74,69,71,66,68,63,65,60,62,57,59,54,56,51,53,48,50,45,47,42,44,39,41,36,38,33,35,30,32,27,29,24,26,21,23,18,20,15,17,12,14,9,11,6,8,3,5)(84,85,86)(87,88,89,91,93,95,97,99,101,103,105,107,109,111,110,108,106,104,102,100,98,96,94,92,90)(112,113,115,118)(114,117,119,116), (1,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,79,73,67,61,55,49,43,37,31,25,19,13,7,2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83,82,76,70,64,58,52,46,40,34,28,22,16,10,4)(84,85,86)(88,90)(89,92)(91,94)(93,96)(95,98)(97,100)(99,102)(101,104)(103,106)(105,108)(107,110)(109,111)(112,114,115,119)(113,116,118,117) );

# 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(83195276699457020363312626218078054384593940819372001001375549399731547,49800); a := GPC.1; b := GPC.2;

# Define the group as a permutation group: 
Group( (1,2,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82,81,83,78,80,75,77,72,74,69,71,66,68,63,65,60,62,57,59,54,56,51,53,48,50,45,47,42,44,39,41,36,38,33,35,30,32,27,29,24,26,21,23,18,20,15,17,12,14,9,11,6,8,3,5)(84,85,86)(87,88,89,91,93,95,97,99,101,103,105,107,109,111,110,108,106,104,102,100,98,96,94,92,90)(112,113,115,118)(114,117,119,116), (1,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,79,73,67,61,55,49,43,37,31,25,19,13,7,2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83,82,76,70,64,58,52,46,40,34,28,22,16,10,4)(84,85,86)(88,90)(89,92)(91,94)(93,96)(95,98)(97,100)(99,102)(101,104)(103,106)(105,108)(107,110)(109,111)(112,114,115,119)(113,116,118,117) );

# Define the group as a matrix group with coefficients in GLFp: 
Group([[[ Z(499)^148, Z(499)^90 ], [ Z(499)^338, Z(499)^397 ]], [[ Z(499)^428, Z(499)^2 ], [ Z(499), Z(499)^428 ]]]);

# 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