# Oscar code for working with abstract group 311040.j.
# If you have not already loaded the Oscar package, you should type "using Oscar;" before running the code below.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G = @permutation_group(81, (1,56,49,12,42,35)(2,3,48)(4,18,78,21,81,36)(5,8,34,19,33,9)(6,25,44,38,39,69)(7,58,11)(10,14,67,53,20,75)(13,61,51,74,23,59)(15,27,32)(16,57,66,30,31,17)(22,55,65,72,70,40)(24,73,62,54,52,76)(26,60,29,77,47,41)(28,46,37,64,68,63)(43,71,45,80,50,79), (1,78,9,48,64,43,52,53)(2,3,4,10,19,81,62,49)(5,65,80,39,70,72,58,33)(6,30,37,28,77,71,40,60)(7,22,44,34,57,41,55,68)(8,46,75,14,45,67,24,50)(11,17)(13,36,21,15,35,31,27,74)(16,18,26,69,59,76,23,32)(25,56,61,38,66,47,73,54)(29,51)(42,63))

# Order of the group: 
order(G)

# Exponent of the group: 
exponent(G)

# Automorphism group: 
automorphism_group(G)

# Composition factors of the group: 
composition_series(G)

# Nilpotency class of the group: 
if is_nilpotent(G) nilpotency_class(G) end

# Derived length of the group: 
derived_length(G)

# Determine if the group G is abelian: 
is_abelian(G)

# Determine if the group G is cyclic: 
is_cyclic(G)

# Determine if the group G is elementary abelian: 
is_elementary_abelian(G)

# Determine if the group G is nilpotent: 
is_nilpotent(G)

# Determine if the group G is perfect: 
is_perfect(G)

# Determine if the group G is a p-group: 
is_pgroup(G)

# Determine if the group G is simple: 
is_simple(G)

# Determine if the group G is solvable: 
is_solvable(G)

# Determine if the group G is supersolvable: 
is_supersolvable(G)

# Compute statistics for the group G: 
# Oscar code to output the first two rows of the group statistics table
element_orders = [order(g) for g in elements(G)]
orders = sort(unique(element_orders))
println("Orders: ", orders)
element_counts = [count(==(n), element_orders) for n in orders]
println("Elements: ", element_counts, " ", order(G))
ccs = conjugacy_classes(G)
cc_orders = [order(representative(cc)) for cc in ccs]
cc_counts = [count(==(n), cc_orders) for n in orders]
println("Conjugacy classes: ", cc_counts, " ", length(ccs))

# List of conjugacy classes of the group: 
conjugacy_classes(G)  # Output not guaranteed to exactly match the LMFDB table

# Compute statistics about the characters of G: 
# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities.
character_degrees(G)

# Define the group as a permutation group: 
@permutation_group(81, (1,56,49,12,42,35)(2,3,48)(4,18,78,21,81,36)(5,8,34,19,33,9)(6,25,44,38,39,69)(7,58,11)(10,14,67,53,20,75)(13,61,51,74,23,59)(15,27,32)(16,57,66,30,31,17)(22,55,65,72,70,40)(24,73,62,54,52,76)(26,60,29,77,47,41)(28,46,37,64,68,63)(43,71,45,80,50,79), (1,78,9,48,64,43,52,53)(2,3,4,10,19,81,62,49)(5,65,80,39,70,72,58,33)(6,30,37,28,77,71,40,60)(7,22,44,34,57,41,55,68)(8,46,75,14,45,67,24,50)(11,17)(13,36,21,15,35,31,27,74)(16,18,26,69,59,76,23,32)(25,56,61,38,66,47,73,54)(29,51)(42,63))

# The primary decomposition of the group: 
abelian_invariants(G)

# The abelianization of the group: 
quo(G, derived_subgroup(G)[1])

# List of subgroups of the group: 
subgroups(G)

# Center of the group: 
center(G)

# Commutator subgroup of the group G: 
derived_subgroup(G)

# Frattini subgroup of the group G: 
frattini_subgroup(G)

# Fitting subgroup of the group G: 
fitting_subgroup(G)

# Radical of the group G: 
solvable_radical(G)

# Socle of the group G: 
socle(G)

# Derived series of the group G: 
derived_series(G)

# Chief series of the group G: 
chief_series(G)

# The lower central series of the group G: 
lower_central_series(G)

# The upper central series of the group G: 
upper_central_series(G)

# Character table: 
character_table(G)  # Output not guaranteed to exactly match the LMFDB table