# SageMath code for working with abstract group 5668704.s.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(6030651065619853294650447015658583710334957676522772022662910296208966270471106052347872441904944045879806124745730624628125854934390562882851475528523841279238215995652279499384086261685707714096984122330096258358489133073365038882648215758245275349039830782295887996150321722233082263640687622325709788183662002938266788146140426413448390006353151269589125708626340228644061395214921578989954103689635404800936981383315355529293184529309609872757978111574509980173012225894816963465861189471504613096329256315647,5668704)'); a = GPC.1; b = GPC.2; c = GPC.4; d = GPC.6; e = GPC.8; f = GPC.10; g = GPC.11; h = GPC.12; i = GPC.13; j = GPC.14; k = GPC.15; l = GPC.16;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 54681)

# 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