# SageMath code for working with abstract group 1285608.a.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G = PSL(2,137)

# 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 group as Lie representations: 
PSL(2,137)
PSU(2,137)

# Define the group as a permutation group: 
PermutationGroup(['(1,71,2)(3,86,124)(4,7,10)(5,132,24)(6,41,116)(8,36,111)(9,128,20)(11,28,66)(12,81,13)(14,88,84)(15,49,50)(16,125,104)(17,56,129)(18,109,70)(19,133,76)(21,101,126)(22,127,60)(23,96,135)(25,130,92)(26,51,131)(27,136,48)(29,69,100)(30,35,113)(31,89,105)(32,114,78)(33,55,42)(34,61,65)(37,45,98)(38,120,74)(39,117,122)(40,77,57)(43,134,82)(44,85,94)(46,90,80)(47,63,121)(52,83,123)(53,59,79)(54,107,115)(58,67,108)(62,106,72)(64,138,68)(73,93,99)(75,112,95)(87,91,118)(97,119,110)(102,103,137)', '(3,123,111,7,76,71,85,95,92,77,124,28,14,102,22,16,118,59,66,94,11,34,51,83,44,52,54,117,116,29,50,121,96,41,109,79,72,68,128,86,37,35,81,103,129,26,67,132,43,120,131,105,78,40,19,57,6,80,27,133,136,137,126,33,99,53,110,48)(4,5,8,114,61,135,84,122,101,63,36,10,21,98,9,74,115,12,38,60,106,104,55,13,73,69,62,32,100,45,20,91,112,25,24,87,89,97,58,90,107,130,47,75,82,23,125,119,39,127,113,17,64,49,46,56,70,65,134,30,18,138,93,31,88,42,108,15)'])

# 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