# Oscar code for working with abstract group 113460.a.
# 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(62, (1,62,2)(3,32,61)(4,14,59)(5,50,60)(6,28,29)(7,18,11)(8,21,47)(9,27,25)(10,41,15)(12,44,24)(13,33,38)(16,22,30)(17,43,56)(20,52,40)(23,54,49)(26,31,51)(34,42,48)(35,36,58)(37,55,39)(46,57,53), (3,39,14,6,7,20,47,28,17,25,42,24,34,57,50,4,11,22,52,16,55,21,35,56,12,46,36,5,27,32)(8,31,41,23,40,48,37,18,45,58,59,51,26,62,33,38,60,29,19,53,9,30,44,10,49,13,43,54,61,15))

# 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 group as Lie representations: 
matrix_group([matrix(GF(61), [[60, 1], [60, 0]]), matrix(GF(61), [[2, 0], [0, 31]])])

# Define the group as a permutation group: 
@permutation_group(62, (1,62,2)(3,32,61)(4,14,59)(5,50,60)(6,28,29)(7,18,11)(8,21,47)(9,27,25)(10,41,15)(12,44,24)(13,33,38)(16,22,30)(17,43,56)(20,52,40)(23,54,49)(26,31,51)(34,42,48)(35,36,58)(37,55,39)(46,57,53), (3,39,14,6,7,20,47,28,17,25,42,24,34,57,50,4,11,22,52,16,55,21,35,56,12,46,36,5,27,32)(8,31,41,23,40,48,37,18,45,58,59,51,26,62,33,38,60,29,19,53,9,30,44,10,49,13,43,54,61,15))

# 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