# Oscar code for working with abstract group 5668704.it.
# 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(36, (1,15,27,3,13,25,2,14,26)(4,12,18,23,30,34,5,10,17,22,29,35,6,11,16,24,28,36)(7,19,31,8,21,32,9,20,33), (1,25)(2,27)(3,26)(4,17,29,5,16,28,6,18,30)(7,19)(8,21)(9,20)(10,22,34,11,24,35,12,23,36)(13,15,14)(31,33,32), (1,16,20,35,3,18,19,36,2,17,21,34)(4,8,22,26,5,9,24,27,6,7,23,25)(10,14,29,31,11,15,28,32,12,13,30,33))

# 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(36, (1,15,27,3,13,25,2,14,26)(4,12,18,23,30,34,5,10,17,22,29,35,6,11,16,24,28,36)(7,19,31,8,21,32,9,20,33), (1,25)(2,27)(3,26)(4,17,29,5,16,28,6,18,30)(7,19)(8,21)(9,20)(10,22,34,11,24,35,12,23,36)(13,15,14)(31,33,32), (1,16,20,35,3,18,19,36,2,17,21,34)(4,8,22,26,5,9,24,27,6,7,23,25)(10,14,29,31,11,15,28,32,12,13,30,33))

# Define the group from the transitive group database: 
transitive_group(36, 54911)

# 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