# Gap code for working with abstract group 60466176.ev.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(1454603635616177113005173519290108820391766430170881026027375863652606206079691730978011248183195896184445907498092504694324252020608525956760274246277634268658770678846502074412912101566137343083550697053752218014671436175807033561899222487636889855433754459678343877998825702835755422326315313903262575959210555628201682918755466009503427433969452894430382854233290699268742707468430245111571072400106178438026362342951171425977802269091463226333113326966353813780552235557326498258315964518531096247621241682036393246038268748197099467123577975483070880515608283400052767528891296757463472422367341966688601831431146531953731561154779074022752883556501049829778324495707861913922773108030280844169499808930449086993016644256767777530594559595660017610292113913717024666809726630671986510613801820014114308675090192574942182480158767394658555531706761529180376770422377566076073629438789434586854739797210918643232475036103999598829576534663209248122869700511676597080396394245626133753144006889342947242130105099676214085590284572399192891888929273863308117088222574744728638496874671709033531387549338754050108425872059546378631035750227398650129560917985349652822050456544269287696835373754589230495374774124184299973772330543391359,60466176); a := GPC.1; b := GPC.2; c := GPC.4; d := GPC.6; e := GPC.8; f := GPC.10; g := GPC.12; h := GPC.15; i := GPC.16; j := GPC.17; k := GPC.18; l := GPC.19; m := GPC.20;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 74854);

# 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