# Oscar code for working with abstract group 124500.c.
# 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).


# Define group as a dicyclic group: 
G = dicyclic_group(124500)

# 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(215, (2,4)(3,5)(6,11)(7,12)(8,10)(9,13)(14,22)(15,23)(16,21)(17,24)(18,20)(19,25)(26,30)(27,31)(28,32)(29,33)(34,38)(35,39)(36,40)(37,41)(42,46)(43,47)(44,48)(45,49)(50,54)(51,55)(52,56)(53,57)(58,62)(59,63)(60,64)(61,65)(66,70)(67,71)(68,72)(69,73)(74,78)(75,79)(76,80)(77,81)(82,86)(83,87)(84,88)(85,89)(90,94)(91,95)(92,96)(93,97)(98,102)(99,103)(100,104)(101,105)(106,110)(107,111)(108,112)(109,113)(114,118)(115,119)(116,120)(117,121)(122,125)(123,124)(126,127,128,129)(130,131,133,136,140,145,151,157,163,169,175,181,187,193,198,168,139,143,148,154,160,166,172,178,184,190,196,201,205,208,210,186,156,132,134,137,141,146,152,158,164,170,176,182,188,194,199,203,174,144,149,155,161,167,173,179,185,191,197,202,206,209,211,212,192,162,135,138,142,147,153,159,165,171,177,183,189,195,200,204,207,180,150)(213,214,215), (1,2,6,14,19,28,35,42,45,52,59,66,69,76,83,90,93,100,107,114,117,124,120,113,110,103,96,89,86,79,72,65,62,55,48,41,38,31,24,13,18,21,10,3,7,15,26,29,36,43,50,53,60,67,74,77,84,91,98,101,108,115,122,125,119,112,105,102,95,88,81,78,71,64,57,54,47,40,33,30,23,12,5,8,16,20,9,17,27,34,37,44,51,58,61,68,75,82,85,92,99,106,109,116,123,121,118,111,104,97,94,87,80,73,70,63,56,49,46,39,32,25,22,11,4)(126,128)(127,129)(130,132,135,139,144,150,156,162,168,174,180,186,192,198,203,207,210,212,193,199,204,208,211,187,194,200,205,209,181,188,195,201,206,175,182,189,196,202,169,176,183,190,197,163,170,177,184,191,157,164,171,178,185,151,158,165,172,179,145,152,159,166,173,140,146,153,160,167,136,141,147,154,161,133,137,142,148,155,131,134,138,143,149)(213,214,215))

# Define the group as a matrix group with coefficients in GLFp: 
matrix_group([matrix(GF(499), [[436, 74], [132, 63]]), matrix(GF(499), [[345, 162], [237, 345]])])

# 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