# Gap code for working with abstract group 498000.a.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G := Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83)(84,85,86,87,89,91,88,90)(92,93,95,97,99,101,103,105,107,124,147,152,155,157,159,161,163,165,167,171,170,191,196,199,201,203,205,207,209,210,208,206,204,202,200,197,198,195,172,168,166,164,162,160,158,156,153,154,151,126,123,121,119,117,115,113,111,109,110,112,114,116,118,120,122,125,133,129,127,131,135,137,139,141,143,145,149,148,169,174,177,179,181,183,185,187,189,193,192,211,213,216,214,215,212,194,190,188,186,184,182,180,178,175,176,173,150,146,144,142,140,138,136,132,128,130,134,108,106,104,102,100,98,96,94)(217,218,219), (85,87)(86,88)(90,91)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,127)(110,129)(111,131)(112,133)(113,135)(114,125)(115,137)(116,122)(117,139)(118,120)(119,141)(121,143)(123,145)(124,134)(126,149)(128,152)(130,147)(132,155)(136,157)(138,159)(140,161)(142,163)(144,165)(146,167)(148,151)(150,171)(153,174)(154,169)(156,177)(158,179)(160,181)(162,183)(164,185)(166,187)(168,189)(170,173)(172,193)(175,196)(176,191)(178,199)(180,201)(182,203)(184,205)(186,207)(188,209)(190,210)(192,195)(194,208)(197,213)(198,211)(200,216)(202,214)(204,215)(206,212) );

# 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(2058369659322288639891289340025748011726327651723498302114540073455150195789258215504410954177634949,498000); a := GPC.1; b := GPC.4;

# Define the group as a permutation group: 
Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83)(84,85,86,87,89,91,88,90)(92,93,95,97,99,101,103,105,107,124,147,152,155,157,159,161,163,165,167,171,170,191,196,199,201,203,205,207,209,210,208,206,204,202,200,197,198,195,172,168,166,164,162,160,158,156,153,154,151,126,123,121,119,117,115,113,111,109,110,112,114,116,118,120,122,125,133,129,127,131,135,137,139,141,143,145,149,148,169,174,177,179,181,183,185,187,189,193,192,211,213,216,214,215,212,194,190,188,186,184,182,180,178,175,176,173,150,146,144,142,140,138,136,132,128,130,134,108,106,104,102,100,98,96,94)(217,218,219), (85,87)(86,88)(90,91)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,127)(110,129)(111,131)(112,133)(113,135)(114,125)(115,137)(116,122)(117,139)(118,120)(119,141)(121,143)(123,145)(124,134)(126,149)(128,152)(130,147)(132,155)(136,157)(138,159)(140,161)(142,163)(144,165)(146,167)(148,151)(150,171)(153,174)(154,169)(156,177)(158,179)(160,181)(162,183)(164,185)(166,187)(168,189)(170,173)(172,193)(175,196)(176,191)(178,199)(180,201)(182,203)(184,205)(186,207)(188,209)(190,210)(192,195)(194,208)(197,213)(198,211)(200,216)(202,214)(204,215)(206,212) );

# Define the group as a matrix group with coefficients in GLFp: 
Group([[[ Z(499)^414, Z(499)^56 ], [ Z(499)^55, Z(499)^414 ]], [[ Z(499)^0, 0*Z(499) ], [ 0*Z(499), Z(499)^249 ]]]);

# 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