# SageMath code for working with abstract group 12754584.he.
# Some of these functions may take a long time to execute (this depends on the group).


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

# 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(7607668701501790573228704488299130684201601382681422510130056759695503695344359501807017776805664970368239242748541071765708022592683821227963326771251689263950585292364300747858324200378802162716213301423809005057068862975694648335565414220675386825158619621512247925886349797328633954954596017156815174658617407032632872498863211574064122609497935292876535094403106113930457907881438685465308635022669666436955798726763801748233791618412957231027782261518437823256799630550897849677177629655484840446651589896950078613324207613210189889507519507255699777270962661034045343769647625778397630218089528992012382112417730401831032151375701150659007286111555769358997106983142727283457002612010651212088112529531612712988139784185975908351,12754584)'); a = GPC.1; b = GPC.2; c = GPC.4; d = GPC.6; e = GPC.7; f = GPC.8; g = GPC.9; h = GPC.11; i = GPC.12; j = GPC.14; k = GPC.15; l = GPC.16;

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

# Define the group from the transitive group database: 
TransitiveGroup(36, 62083)

# 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