# SageMath code for working with abstract group 11664.kv.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(3736206736914572732371176538338573298327729471332021339675670395103035095875381889405422227048755497135150582706638795111865356981490165501023,11664)'); a = GPC.1; b = GPC.2; c = GPC.4; d = GPC.5; e = GPC.7; f = GPC.8; g = GPC.9; h = GPC.10;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 9560)

# 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