# Oscar code for working with abstract group 699840.c.
# 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(54, (1,4,30,27,11,3,13,10)(2,19,29,54,14,53,8,52)(5,50)(7,44)(9,43,24,46,36,41,25,42)(12,39,16,37,28,49,17,18)(15,51,32,40,31,34,33,38)(20,23,48,47,22,35,45,21), (1,54,37,24,35,33,10,41,51,36,45,17,3,53,39,8,47,12)(2,48,31,13,52,18,7,23,15,26,19,49,14,21,5,30,43,34)(4,44,50,25,6,28,27,42,38,29,22,16,11,46,40,9,20,32), (1,38,17,3,50,5)(2,20,52,29,21,41)(4,49,33,30,37,31)(6,44,36,23,42,8)(7,47,43,25,45,53)(9,22,54,14,48,19)(10,40,16)(11,18,15,13,51,12)(24,35,46)(26,39,32,27,34,28))

# 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(54, (1,4,30,27,11,3,13,10)(2,19,29,54,14,53,8,52)(5,50)(7,44)(9,43,24,46,36,41,25,42)(12,39,16,37,28,49,17,18)(15,51,32,40,31,34,33,38)(20,23,48,47,22,35,45,21), (1,54,37,24,35,33,10,41,51,36,45,17,3,53,39,8,47,12)(2,48,31,13,52,18,7,23,15,26,19,49,14,21,5,30,43,34)(4,44,50,25,6,28,27,42,38,29,22,16,11,46,40,9,20,32), (1,38,17,3,50,5)(2,20,52,29,21,41)(4,49,33,30,37,31)(6,44,36,23,42,8)(7,47,43,25,45,53)(9,22,54,14,48,19)(10,40,16)(11,18,15,13,51,12)(24,35,46)(26,39,32,27,34,28))

# 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