# Oscar code for working with abstract group 216.155.
# 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 = small_group(216, 155)

# 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(12, (2,4,5,7,6,8,9,3)(11,12), (2,5,6,9)(3,4,7,8), (2,6)(3,7)(4,8)(5,9), (1,2,6)(3,4,5)(7,9,8), (1,3,7)(2,4,9)(5,8,6), (10,11,12))

# Define the group as a matrix group with coefficients in GLFp: 
matrix_group([matrix(GF(3), [[2, 0, 0, 1], [0, 1, 0, 0], [1, 0, 1, 1], [2, 0, 0, 0]]), matrix(GF(3), [[1, 1, 2, 2], [0, 2, 0, 0], [0, 1, 1, 0], [0, 2, 1, 2]]), matrix(GF(3), [[2, 1, 1, 2], [2, 1, 0, 2], [1, 1, 2, 2], [2, 2, 2, 2]]), matrix(GF(3), [[0, 2, 2, 1], [2, 0, 1, 0], [2, 0, 1, 2], [2, 1, 1, 1]]), matrix(GF(3), [[0, 1, 2, 1], [2, 1, 2, 1], [2, 0, 1, 2], [1, 2, 1, 0]]), matrix(GF(3), [[1, 0, 2, 2], [2, 1, 0, 2], [1, 0, 0, 0], [0, 0, 1, 2]])])

# Define the group from the transitive group database: 
transitive_group(24, 568)
transitive_group(27, 80)
transitive_group(36, 288)

# 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