# Gap code for working with abstract group 7558272.oj.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(2310500515322527555419541027251375324848069769593715572814888036950183125073946897564254701835643861404546119811361173688926233622261700673176268786356139184495866029523754587045363197413658924783406579930070512739615935841110419980389362615515809055508816545994637694339238018677933444511557205830776624122455617715859052795791690069025211638252778868738873009672782427478093992066739022594715272692257246625720706403422409038605201418378944437889069157418640699476405729827030575780354933708530793086873277677172173945854748858227039171475393703908390924083174412344660126082213579683077994239818697131636916284788321095934973826964139970897914478676925493288504257042772553616974825790275475800321717866382681194526872252605630994631217425845743370011074091431503362655231,7558272); a := GPC.1; b := GPC.2; c := GPC.4; d := GPC.6; e := GPC.7; f := GPC.9; g := GPC.10; h := GPC.12; i := GPC.13; j := GPC.14; k := GPC.15; l := GPC.16; m := GPC.17;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 57840);

# 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