# SageMath code for working with abstract group 729.395.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(5331097505850465300,729)'); a = GPC.1; b = GPC.2; c = GPC.3; d = GPC.5;

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

# Define the group as a matrix group with coefficients in GLZq: 
MS = MatrixSpace(Integers(81), 2, 2)
MatrixGroup([MS([[28, 0], [0, 1]]), MS([[1, 45], [54, 55]]), MS([[22, 20], [3, 4]]), MS([[28, 0], [0, 28]]), MS([[1, 27], [0, 1]]), MS([[46, 0], [0, 64]])])

# 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