# SageMath code for working with abstract group 384.4091.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G = PermutationGroup(['(2,6,8,15)(3,10)(4,11)(7,13,16,9)(12,14)(18,19)', '(1,2)(3,7)(4,13)(5,8)(6,14)(9,11)(10,15)(12,16)', '(1,3,11,14,5,12,4,10)(2,7,9,6,8,16,13,15)(18,19)', '(1,3,4,10,5,12,11,14)(2,6,9,16,8,15,13,7)', '(2,8)(6,15)(7,16)(9,13)', '(1,4,5,11)(2,9,8,13)(3,10,12,14)(6,16,15,7)', '(1,5)(2,8)(3,12)(4,11)(6,15)(7,16)(9,13)(10,14)', '(17,18,19)'])

# 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(53366788838315611669453438260807052937800245476865,384)'); a = GPC.1; b = GPC.2; c = GPC.4;

# Define the group as a permutation group: 
PermutationGroup(['(2,6,8,15)(3,10)(4,11)(7,13,16,9)(12,14)(18,19)', '(1,2)(3,7)(4,13)(5,8)(6,14)(9,11)(10,15)(12,16)', '(1,3,11,14,5,12,4,10)(2,7,9,6,8,16,13,15)(18,19)', '(1,3,4,10,5,12,11,14)(2,6,9,16,8,15,13,7)', '(2,8)(6,15)(7,16)(9,13)', '(1,4,5,11)(2,9,8,13)(3,10,12,14)(6,16,15,7)', '(1,5)(2,8)(3,12)(4,11)(6,15)(7,16)(9,13)(10,14)', '(17,18,19)'])

# Define the group as a matrix group with coefficients in GLZN: 
MS = MatrixSpace(Integers(48), 2, 2)
MatrixGroup([MS([[35, 0], [0, 1]]), MS([[1, 6], [0, 1]]), MS([[1, 24], [0, 1]]), MS([[1, 3], [0, 1]]), MS([[1, 16], [0, 1]]), MS([[25, 0], [0, 1]]), MS([[1, 12], [0, 1]]), MS([[19, 11], [0, 47]])])

# 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