# SageMath code for working with abstract group 1296.855.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(8678127665056083331491148876880960964479162285532018538256505891385,1296)'); a = GPC.1; b = GPC.2; c = GPC.4; d = GPC.6;

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

# 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