# SageMath code for working with abstract group 708588.je.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(5324324220548890227568362481803448504198505464973787969251609987818393940176582529566684745927951727331997998909823853454180866694044748232656276268848259378110325960142944503453819033610728533435356860514379892511288913820113131852921861510520279881171566680300964373953932567057927853484354314824703,708588)'); a = GPC.1; b = GPC.3; c = GPC.5; d = GPC.6; e = GPC.8; f = GPC.9; g = GPC.10; h = GPC.11; i = GPC.12; j = GPC.13;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 34087)

# 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