# SageMath code for working with abstract group 124500.a.
# Some of these functions may take a long time to execute (this depends on the group).


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

# Order of the group: 
G.order()

# Exponent of the group: 
G.exponent()

# Automorphism group: 
libgap(G).AutomorphismGroup()

# Composition factors of the group: 
G.composition_series()

# Nilpotency class of the group: 
libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1

# Derived length of the group: 
libgap(G).DerivedLength()

# Determine if the group G is abelian: 
G.is_abelian()

# Determine if the group G is cyclic: 
G.is_cyclic()

# Determine if the group G is elementary abelian: 
G.is_elementary_abelian()

# Determine if the group G is nilpotent: 
G.is_nilpotent()

# Determine if the group G is perfect: 
G.is_perfect()

# Determine if the group G is a p-group: 
G.is_pgroup()

# Determine if the group G is polycyclic: 
G.is_polycyclic()

# Determine if the group G is simple: 
G.is_simple()

# Determine if the group G is solvable: 
G.is_solvable()

# Determine if the group G is supersolvable: 
G.is_supersolvable()

# Compute statistics for the group G: 
# Sage code to output the first two rows of the group statistics table
element_orders = [g.order() for g in G]
orders = sorted(list(set(element_orders)))
print("Orders:", orders)
print("Elements:", [element_orders.count(n) for n in orders], G.order())
cc_orders = [cc[0].order() for cc in G.conjugacy_classes()]
print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

# List of conjugacy classes of the group: 
G.conjugacy_classes()  # Output not guaranteed to exactly match the LMFDB table

# Compute statistics about the characters of G: 
# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i
character_degrees = [c[0] for c in G.character_table()]
[[n, character_degrees.count(n)] for n in set(character_degrees)]

# Define the group with the given generators and relations: 
# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups
GPC = gap.new('PcGroupCode(6372324679266810872284735140182647424210801730510068964527728319338107,124500)'); a = GPC.1; b = GPC.2;

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

# Define the group as a matrix group with coefficients in GLFp: 
MS = MatrixSpace(GF(499), 2, 2)
MatrixGroup([MS([[1, 0], [0, 498]]), MS([[345, 162], [237, 345]])])

# The abelianization of the group: 
G.quotient(G.commutator())

# The Schur multiplier of the group: 
G.homology(2)

# List of subgroups of the group: 
G.subgroups()

# Center of the group: 
G.center()

# Commutator subgroup of the group G: 
G.commutator()

# Frattini subgroup of the group G: 
G.frattini_subgroup()

# Fitting subgroup of the group G: 
G.fitting_subgroup()

# Socle of the group G: 
G.socle()

# Derived series of the group G: 
G.derived_series()

# Chief series of the group G: 
libgap(G).ChiefSeries()

# The lower central series of the group G: 
G.lower_central_series()

# The upper central series of the group G: 
G.upper_central_series()

# Character table: 
G.character_table()  # Output not guaranteed to exactly match the LMFDB table