# SageMath code for working with abstract group 311040.j.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G = PermutationGroup(['(1,56,49,12,42,35)(2,3,48)(4,18,78,21,81,36)(5,8,34,19,33,9)(6,25,44,38,39,69)(7,58,11)(10,14,67,53,20,75)(13,61,51,74,23,59)(15,27,32)(16,57,66,30,31,17)(22,55,65,72,70,40)(24,73,62,54,52,76)(26,60,29,77,47,41)(28,46,37,64,68,63)(43,71,45,80,50,79)', '(1,78,9,48,64,43,52,53)(2,3,4,10,19,81,62,49)(5,65,80,39,70,72,58,33)(6,30,37,28,77,71,40,60)(7,22,44,34,57,41,55,68)(8,46,75,14,45,67,24,50)(11,17)(13,36,21,15,35,31,27,74)(16,18,26,69,59,76,23,32)(25,56,61,38,66,47,73,54)(29,51)(42,63)'])

# 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 as a permutation group: 
PermutationGroup(['(1,56,49,12,42,35)(2,3,48)(4,18,78,21,81,36)(5,8,34,19,33,9)(6,25,44,38,39,69)(7,58,11)(10,14,67,53,20,75)(13,61,51,74,23,59)(15,27,32)(16,57,66,30,31,17)(22,55,65,72,70,40)(24,73,62,54,52,76)(26,60,29,77,47,41)(28,46,37,64,68,63)(43,71,45,80,50,79)', '(1,78,9,48,64,43,52,53)(2,3,4,10,19,81,62,49)(5,65,80,39,70,72,58,33)(6,30,37,28,77,71,40,60)(7,22,44,34,57,41,55,68)(8,46,75,14,45,67,24,50)(11,17)(13,36,21,15,35,31,27,74)(16,18,26,69,59,76,23,32)(25,56,61,38,66,47,73,54)(29,51)(42,63)'])

# 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