# SageMath code for working with abstract group 1944.3477.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(188540410192601772639868559848555330214882233115193317271667487,1944)'); a = GPC.1; b = GPC.4; c = GPC.5; d = GPC.6; e = GPC.7; f = GPC.8;

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

# 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