# Gap code for working with abstract group 2717908992.l.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(66340334841410460419283549955900856378164807995281071939873311375423692859608787961950071510502875735131584318245679087590071501986203941214725877417074811246332456606072491500462097934573411708302090169065091693004036690095948534103204500749562624504664401761056758346263905035017538765694668754604896773749795110251097427915622194027214290169485411264830835183094683065229493088015107748916241335995819644786476298951837488998898090159240412605240098650646223170625467886705414704131261445936084026354643779958168403403192977034483808828730609492457678545471299795240101825189064951839381001417237366792554056893082875536580902302751484187610403146799034943121893248814317469799185787663636312539436633221012058644375147540414602380268472799777563870210012656541397156764541250126376180346868756827869690691743437337850616146387185522978547774749536342794311912047203102649700453206300609205361021133922945338580727207569817048153709269350372841231918159821816510631276371241002115544870151171713655136058763908498116694158773537393383586154124838300818956394564196770396042220872779487235394590060753586368045453344294668226164062932523393268085550745164177934248759743454680537027272856252828168033555765089005156563788756856866655362266657517931270855596007401269417839307481553792017303777184508772966784817399585172414806259827330525961855529310651255605337169273943021287618501193631114455014235782579836268620240109260606911923972343948904563043465031293383820705754966896178484783702139713506146514202080077648150044707100104035998225540969042622719114691682782118030481997367638020770699449265327111699207207337940806810659468180267609830082401198538855274965847725160079900513412073658834663347176680622113546620791648513878414066494238359042106722482347426299243728233801610550440435034243733602370685677510407298368680444108284599393576025626785144094108486707934007900671925077790918937065577932912770908679508195634156628964900631136193674404719062455870847938743993077636940720663052648076439634362931263127363194814261667716349101091877407503190561708688130462294510658308170978294876049605319945015301872335311640626817273764831271773942133730115185716513121196884561035836419815349703804137567239761237092411407966991452741352072866206413046177768544833983343406939025629058516774565919697336345358991310765526044901195585014075038964763027703486438518793480951187400321117909501894050339366754665588668486940564674683415621489023408795592613131260833387129212924543094610263343707820178349766477796930493240449389663144748202794174740325161748213518154204050975244678347463308101999962611759853728916916352617564706432105941115026301171905661154160074643638566890742035660990074972426677856413646315432757712738938761409419949123565964503995247171189526402855247791389574642349080255214955489572930314022009062215523084737644649958016135432007857953382118406619965061766522023092932057333283277271856764471738577783701504,2717908992); a := GPC.1; b := GPC.3; c := GPC.4; d := GPC.5; e := GPC.7; f := GPC.10; g := GPC.12; h := GPC.14; i := GPC.16; j := GPC.17; k := GPC.18; l := GPC.19; m := GPC.20; n := GPC.21; o := GPC.22; p := GPC.23; q := GPC.24; r := GPC.25; s := GPC.26; t := GPC.27; u := GPC.28; v := GPC.29;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 99464);

# 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