# SageMath code for working with abstract group 5668704.it.
# Some of these functions may take a long time to execute (this depends on the group).


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

# Order of the group: 
G.order()

# Exponent of the group: 
G.exponent()

# Automorphism group: 
libgap(G).AutomorphismGroup()

# Composition factors of the group: 
G.composition_series()

# Nilpotency class of the group: 
libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1

# Derived length of the group: 
libgap(G).DerivedLength()

# Determine if the group G is abelian: 
G.is_abelian()

# Determine if the group G is cyclic: 
G.is_cyclic()

# Determine if the group G is elementary abelian: 
G.is_elementary_abelian()

# Determine if the group G is nilpotent: 
G.is_nilpotent()

# Determine if the group G is perfect: 
G.is_perfect()

# Determine if the group G is a p-group: 
G.is_pgroup()

# Determine if the group G is polycyclic: 
G.is_polycyclic()

# Determine if the group G is simple: 
G.is_simple()

# Determine if the group G is solvable: 
G.is_solvable()

# Determine if the group G is supersolvable: 
G.is_supersolvable()

# Compute statistics for the group G: 
# Sage code to output the first two rows of the group statistics table
element_orders = [g.order() for g in G]
orders = sorted(list(set(element_orders)))
print("Orders:", orders)
print("Elements:", [element_orders.count(n) for n in orders], G.order())
cc_orders = [cc[0].order() for cc in G.conjugacy_classes()]
print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

# List of conjugacy classes of the group: 
G.conjugacy_classes()  # 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
character_degrees = [c[0] for c in G.character_table()]
[[n, character_degrees.count(n)] for n in set(character_degrees)]

# Define the group with the given generators and relations: 
# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups
GPC = gap.new('PcGroupCode(149068683020856106404083804002533493859141235578158705776429979429583197002400994625731224708422422750869493494514378280490244315871588092513449730340497982912247325823095637536193286767821333596573183336784102319994176661073854531578043535442393506142317041696603459820562462219499004735859624180018269884528056144519188729731977461311269098533917519385081274289001483299670753681089242235186923175051322652367677401768934970026719665259140727951831189893518537714985434278704379128042942215155699507201946150687694964531198788999013506938778745265015174435682442671651635268403164301947640850037201752066255540581632531951959539211505272175536659073813596242818376126107222238733101926053785176671319299722301769535251097572411827370097492480194792860415,5668704)'); a = GPC.1; b = GPC.2; c = GPC.4; d = GPC.6; e = GPC.8; f = GPC.10; g = GPC.11; h = GPC.12; i = GPC.13; j = GPC.14; k = GPC.15; l = GPC.16;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 54911)

# The abelianization of the group: 
G.quotient(G.commutator())

# The Schur multiplier of the group: 
G.homology(2)

# List of subgroups of the group: 
G.subgroups()

# Center of the group: 
G.center()

# Commutator subgroup of the group G: 
G.commutator()

# Frattini subgroup of the group G: 
G.frattini_subgroup()

# Fitting subgroup of the group G: 
G.fitting_subgroup()

# Socle of the group G: 
G.socle()

# Derived series of the group G: 
G.derived_series()

# Chief series of the group G: 
libgap(G).ChiefSeries()

# The lower central series of the group G: 
G.lower_central_series()

# The upper central series of the group G: 
G.upper_central_series()

# Character table: 
G.character_table()  # Output not guaranteed to exactly match the LMFDB table