# Oscar code for working with abstract group 432.520.
# 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(432, 520)

# 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(27, (2,3)(4,16,26,21,8,23,14,10)(5,18,27,20,9,22,15,12)(6,17,25,19,7,24,13,11), (2,3)(4,7)(5,9)(6,8)(10,17)(11,16)(12,18)(13,14)(19,23)(20,22)(21,24)(25,26), (4,26,8,14)(5,27,9,15)(6,25,7,13)(10,16,21,23)(11,17,19,24)(12,18,20,22), (4,8)(5,9)(6,7)(10,21)(11,19)(12,20)(13,25)(14,26)(15,27)(16,23)(17,24)(18,22), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18), (1,7,4)(2,8,5)(3,9,6)(10,18,14)(11,16,15)(12,17,13)(19,26,24)(20,27,22)(21,25,23), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26))

# Define the group as a matrix group with coefficients in Z: 
matrix_group([matrix(ZZ, [[-1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0], [-1, 1, 1, 0, 0, 0], [0, 0, -1, -1, -1, 0], [1, -1, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 1]]), matrix(ZZ, [[0, -1, 0, 0, 0, 0], [0, -1, 0, -1, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, -1, 0, -1, 1], [1, 0, 1, 0, 1, -1], [1, -1, 0, 0, 1, -1]])])

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

# Define the group from the transitive group database: 
transitive_group(27, 141)
transitive_group(36, 704)

# 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