# Oscar code for working with abstract group 729.395.
# 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(729, 395)

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

# Define the group as a matrix group with coefficients in GLZq: 
matrix_group([matrix(residue_ring(ZZ, 81)[1][[28, 0], [0, 1]]), matrix(residue_ring(ZZ, 81)[1][[1, 45], [54, 55]]), matrix(residue_ring(ZZ, 81)[1][[22, 20], [3, 4]]), matrix(residue_ring(ZZ, 81)[1][[28, 0], [0, 28]]), matrix(residue_ring(ZZ, 81)[1][[1, 27], [0, 1]]), matrix(residue_ring(ZZ, 81)[1][[46, 0], [0, 64]])])

# 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