# Oscar code for working with abstract group 1285608.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(138, (1,71,2)(3,86,124)(4,7,10)(5,132,24)(6,41,116)(8,36,111)(9,128,20)(11,28,66)(12,81,13)(14,88,84)(15,49,50)(16,125,104)(17,56,129)(18,109,70)(19,133,76)(21,101,126)(22,127,60)(23,96,135)(25,130,92)(26,51,131)(27,136,48)(29,69,100)(30,35,113)(31,89,105)(32,114,78)(33,55,42)(34,61,65)(37,45,98)(38,120,74)(39,117,122)(40,77,57)(43,134,82)(44,85,94)(46,90,80)(47,63,121)(52,83,123)(53,59,79)(54,107,115)(58,67,108)(62,106,72)(64,138,68)(73,93,99)(75,112,95)(87,91,118)(97,119,110)(102,103,137), (3,123,111,7,76,71,85,95,92,77,124,28,14,102,22,16,118,59,66,94,11,34,51,83,44,52,54,117,116,29,50,121,96,41,109,79,72,68,128,86,37,35,81,103,129,26,67,132,43,120,131,105,78,40,19,57,6,80,27,133,136,137,126,33,99,53,110,48)(4,5,8,114,61,135,84,122,101,63,36,10,21,98,9,74,115,12,38,60,106,104,55,13,73,69,62,32,100,45,20,91,112,25,24,87,89,97,58,90,107,130,47,75,82,23,125,119,39,127,113,17,64,49,46,56,70,65,134,30,18,138,93,31,88,42,108,15))

# 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(137), [[136, 1], [136, 0]]), matrix(GF(137), [[3, 0], [0, 46]])])

# Define the group as a permutation group: 
@permutation_group(138, (1,71,2)(3,86,124)(4,7,10)(5,132,24)(6,41,116)(8,36,111)(9,128,20)(11,28,66)(12,81,13)(14,88,84)(15,49,50)(16,125,104)(17,56,129)(18,109,70)(19,133,76)(21,101,126)(22,127,60)(23,96,135)(25,130,92)(26,51,131)(27,136,48)(29,69,100)(30,35,113)(31,89,105)(32,114,78)(33,55,42)(34,61,65)(37,45,98)(38,120,74)(39,117,122)(40,77,57)(43,134,82)(44,85,94)(46,90,80)(47,63,121)(52,83,123)(53,59,79)(54,107,115)(58,67,108)(62,106,72)(64,138,68)(73,93,99)(75,112,95)(87,91,118)(97,119,110)(102,103,137), (3,123,111,7,76,71,85,95,92,77,124,28,14,102,22,16,118,59,66,94,11,34,51,83,44,52,54,117,116,29,50,121,96,41,109,79,72,68,128,86,37,35,81,103,129,26,67,132,43,120,131,105,78,40,19,57,6,80,27,133,136,137,126,33,99,53,110,48)(4,5,8,114,61,135,84,122,101,63,36,10,21,98,9,74,115,12,38,60,106,104,55,13,73,69,62,32,100,45,20,91,112,25,24,87,89,97,58,90,107,130,47,75,82,23,125,119,39,127,113,17,64,49,46,56,70,65,134,30,18,138,93,31,88,42,108,15))

# 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