# Gap code for working with abstract group 12754584.he.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(7607668701501790573228704488299130684201601382681422510130056759695503695344359501807017776805664970368239242748541071765708022592683821227963326771251689263950585292364300747858324200378802162716213301423809005057068862975694648335565414220675386825158619621512247925886349797328633954954596017156815174658617407032632872498863211574064122609497935292876535094403106113930457907881438685465308635022669666436955798726763801748233791618412957231027782261518437823256799630550897849677177629655484840446651589896950078613324207613210189889507519507255699777270962661034045343769647625778397630218089528992012382112417730401831032151375701150659007286111555769358997106983142727283457002612010651212088112529531612712988139784185975908351,12754584); a := GPC.1; b := GPC.2; c := GPC.4; d := GPC.6; e := GPC.7; f := GPC.8; g := GPC.9; h := GPC.11; i := GPC.12; j := GPC.14; k := GPC.15; l := GPC.16;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 62083);

# 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