# Gap code for working with abstract group 34992.ns.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G := Group( (1,5,15)(2,9,23)(3,12,28)(4,14,31)(6,17,22)(7,19,36)(8,21,39)(10,25,35)(11,27,43)(13,30,45)(16,32,38)(18,34,48)(20,37,50)(24,41,47)(26,42,52)(29,44,49)(33,46,54)(40,51,53), (1,4,13)(2,8,20)(5,14,30)(6,16,29)(9,21,37)(12,25,19)(15,31,45)(17,32,44)(22,38,49)(23,39,50)(27,41,34)(28,36,35)(42,51,46)(43,48,47)(52,54,53), (3,10,7)(11,24,18)(12,25,19)(26,40,33)(27,41,34)(28,35,36)(42,51,46)(43,47,48)(52,53,54), (2,6)(5,15)(7,10)(8,16)(9,22)(12,28)(14,31)(17,23)(18,24)(19,35)(20,29)(21,38)(25,36)(27,43)(30,45)(32,39)(33,40)(34,47)(37,49)(41,48)(42,52)(44,50)(46,53)(51,54), (4,8,16)(13,29,20)(14,21,32)(30,44,37)(31,39,38)(45,49,50), (1,3)(2,7,6,10)(4,11,13,26)(5,12)(8,18,29,40)(9,19,17,25)(14,27,30,42)(15,28)(16,24,20,33)(21,34,44,51)(22,35,23,36)(31,43,45,52)(32,41,37,46)(38,47,50,54)(39,48,49,53), (2,6)(8,16)(9,17)(11,26)(18,33)(20,29)(21,32)(22,23)(24,40)(27,42)(34,46)(37,44)(38,39)(41,51)(43,52)(47,53)(48,54)(49,50), (4,13)(7,10)(8,20)(14,30)(16,29)(18,24)(19,25)(21,37)(31,45)(32,44)(33,40)(34,41)(35,36)(38,49)(39,50)(46,51)(47,48)(53,54), (11,24,18)(26,33,40)(27,41,34)(42,46,51)(43,47,48)(52,54,53), (3,11,26)(5,9,17)(7,18,33)(10,24,40)(12,27,42)(14,21,32)(15,22,23)(19,34,46)(25,41,51)(28,43,52)(30,37,44)(31,38,39)(35,47,53)(36,48,54)(45,49,50), (1,2,6)(4,8,16)(5,9,17)(13,20,29)(14,21,32)(15,23,22)(30,37,44)(31,39,38)(45,50,49) );

# 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(3150585384137554458325054523425505931934715843152161797002112631198923994418166956429616714005670600265414657796814933241784884937028805232744408628644345336127,34992); a := GPC.1; b := GPC.2; c := GPC.4; d := GPC.7; e := GPC.8; f := GPC.9; g := GPC.10; h := GPC.11;

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

# 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