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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1280.1116446', 'ambient_counter': 1116446, 'ambient_order': 1280, 'ambient_tex': 'C_2^6:D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 128, 'counter': 12, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '1280.1116446.5.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '5.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 5, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '256.29598', 'subgroup_hash': 29598, 'subgroup_order': 256, 'subgroup_tex': 'C_2^5:D_4', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '1280.1116446', 'aut_centralizer_order': 4, 'aut_label': '5.a1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '1536.408626250', 'aut_weyl_index': 20, 'centralizer': '80.i2', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['10.a1', '10.b1', '10.c1', '10.d1', '10.e1', '10.f1', '10.g1', '10.h1', '10.i1'], 'core': '10.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [8132, -1, 4792, -1], 'generators': [999094447, 12933044760, 14382387365, 14382387360, 11291160, 12945463200, 14374039680, 16], 'label': '1280.1116446.5.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1', 'old_label': '5.a1', 'projective_image': '320.1636', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '5.a1', 'subgroup_fusion': None, 'weyl_group': '16.14'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 168, 'aut_gen_orders': [14, 12], 'aut_gens': [[273313, 321413, 14129, 245383, 262963, 180013, 245615], [321425, 76321, 97079, 107441, 96775, 97063, 76547], [245671, 17575, 262955, 241925, 179725, 180013, 107513]], 'aut_group': None, 'aut_hash': 1860475951476847956, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 88080384, 'aut_permdeg': 72, 'aut_perms': [15409986402238812430922535011996385437504830040922932305549222329780380379875288834644405215645825961312, 24353470394232406435777715449406777896245313427838781858353143510008125060617872819134937121647180478009], 'aut_phi_ratio': 688128.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 1, 8, 1], [2, 2, 56, 1], [2, 8, 2, 1], [4, 8, 14, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^{15}.C_2^4.\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8483965898770251041, 'autcent_nilpotent': True, 'autcent_order': 524288, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{13}.C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '168.42', 'autcentquo_hash': 42, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PSL(2,7)', 'cc_stats': [[1, 1, 1], [2, 1, 15], [2, 2, 56], [2, 8, 2], [4, 8, 14]], 'center_label': '16.14', 'center_order': 16, 'central_product': True, 'central_quotient': '16.14', 'commutator_count': 1, 'commutator_label': '8.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 29598, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['128.1578', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [2, 2, 1, 56], [2, 8, 1, 2], [4, 8, 1, 14]], 'element_repr_type': 'GLZN', 'elementary': 2, 'eulerian_function': 3720, 'exponent': 4, 'exponents_of_order': [8], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '32.51', 'hash': 29598, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 2, 1, 1, 2], 'inner_gens': [[273313, 321413, 14129, 245383, 262963, 180013, 321497], [273313, 321413, 14129, 245383, 262963, 180013, 155597], [273313, 321413, 14129, 245383, 262963, 180013, 245615], [273313, 321413, 14129, 245383, 262963, 180013, 155885], [273313, 321413, 14129, 245383, 262963, 180013, 245615], [273313, 321413, 14129, 245383, 262963, 180013, 245615], [183175, 79511, 14129, 155821, 262963, 180013, 245615]], 'inner_hash': 14, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': False, 'inner_tex': 'C_2^4', 'inner_used': [1, 2, 4, 7], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 56]], 'label': '256.29598', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^5:D4', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 88, 'number_divisions': 88, 'number_normal_subgroups': 943, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 16445, 'number_subgroups': 32623, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 143], [4, 112]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [14, 4], 'outer_gen_pows': [13825, 13825], 'outer_gens': [[262663, 183175, 96791, 324875, 179725, 96775, 107501], [321709, 13829, 14129, 159275, 180013, 262963, 97133]], 'outer_group': None, 'outer_hash': 1790878771140052493, 'outer_nilpotent': False, 'outer_order': 5505024, 'outer_permdeg': 58, 'outer_perms': [901780755758347057444186394308658225432002081719583360966169697751807757125009, 637177896385242399314446046793464066393938767567929912027180532402659639550408], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^9.C_2^6.\\PSL(2,7)', 'pc_rank': 7, 'perfect': False, 'permutation_degree': None, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 56]], 'representations': {'PC': {'code': 4613981833200406592, 'gens': [1, 2, 3, 4, 5, 6, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 12550, 5390, 590, 166]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1083311105, 1087522049, 2346386634, 30819648, 27215168, 2211644610, 1087570945, 22086720]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [14113, 76325, 179725, 321485, 79799, 13829, 14047, 158987]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5:D_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [8, 4, 20, 4, 6], 'aut_gens': [[999094447, 7784985720, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16], [13413219247, 28037364600, 12419280, 12944332800, 12945463200, 1967535360, 14382387376, 14382387374, 16], [999821663, 28026851160, 14382386640, 1959916440, 11291160, 1968262560, 14382387360, 14382387374, 16], [32152263127, 7785752520, 1959916440, 14375134800, 11291160, 12419280, 14382387376, 14382387374, 16], [50433521767, 45328238640, 12419280, 1961009280, 1968262560, 12934540800, 14382387376, 14382387365, 16], [50437151287, 38422108920, 14381661720, 1498320, 12419280, 12945463200, 14382387360, 14, 16]], 'aut_group': None, 'aut_hash': 7528179241671287496, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 30720, 'aut_permdeg': 84, 'aut_perms': [1224214264156756305972459081385868437016672812413093261051798836245605953527311130834386879978729228414342819090170727218848962, 1665158098409651600076695658128423147137656145211922549327732150829630418356405559120628711136527852522774336446031959471675645, 945483814738970861930924384436175117585197562517691177189895402485685563786570481737703925135273769821341690930267327774246182, 1657803086861279899127916395774633352039724712897823092169918528549773853224565741433188739330867895554153677668931198097968356, 2884011085774047757629368874215186148648504920249310572413221762299850625947569756802100770783574167742891351081751004683261162], 'aut_phi_ratio': 60.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 5, 3, 2], [2, 5, 6, 1], [2, 10, 6, 1], [2, 40, 2, 1], [4, 40, 2, 1], [4, 40, 6, 2], [5, 32, 2, 1], [10, 32, 2, 1], [10, 32, 4, 1], [10, 32, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_{16}.C_2^4.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '960.11359', 'autcentquo_hash': 11359, 'autcentquo_nilpotent': False, 'autcentquo_order': 960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{16}:C_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 5, 12], [2, 10, 6], [2, 40, 2], [4, 40, 14], [5, 32, 2], [10, 32, 14]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '320.1636', 'commutator_count': 1, 'commutator_label': '160.235', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1116446, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['640.21467', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 5, 1, 12], [2, 10, 1, 6], [2, 40, 1, 2], [4, 40, 1, 14], [5, 32, 2, 1], [10, 32, 2, 3], [10, 32, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 21420, 'exponent': 20, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '640.21537', 'hash': 1116446, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [2, 5, 2, 2, 2, 2, 1, 2, 1], 'inner_gens': [[999094447, 18855259920, 12934540800, 14375134800, 12944332800, 1967535360, 14382387360, 14382387374, 16], [25644826087, 7784985720, 1967535360, 14381661720, 1961009280, 12934540800, 14382387360, 14382387365, 16], [999821647, 33096632520, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16], [2446940287, 34529564280, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16], [2446940287, 9744860280, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16], [999094447, 34533193800, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16], [999094447, 7784985720, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16], [999094463, 7784985720, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16], [999094447, 7784985720, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16]], 'inner_hash': 1636, 'inner_nilpotent': False, 'inner_order': 320, 'inner_split': False, 'inner_tex': 'C_2\\wr D_5', 'inner_used': [1, 2, 3, 8], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 18], [5, 24], [10, 6]], 'label': '1280.1116446', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2^6:D10', 'ngens': 9, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 56, 'number_divisions': 46, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 438, 'number_subgroup_classes': 4827, 'number_subgroups': 46862, 'old_label': None, 'order': 1280, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 207], [4, 560], [5, 64], [10, 448]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 6, 4, 2], 'outer_gen_pows': [999094447, 14382387365, 0, 0], 'outer_gens': [[999094447, 45316590480, 14381661720, 12419280, 1961009280, 1968262560, 14382387376, 5, 16], [14861792287, 7784985720, 14375134800, 11291160, 12419280, 1968262560, 14382387376, 14382387365, 16], [999094447, 7784985720, 12933044760, 12945463200, 14374039680, 1967535360, 14382387376, 14, 16], [14861792287, 7784985720, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16]], 'outer_group': '96.210', 'outer_hash': 210, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 13, 'outer_perms': [3437250487, 3125253630, 1086190560, 1611308161], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4:D_6', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 4], [5, 24], [8, 2], [10, 6]], 'representations': {'PC': {'code': '3926868250457796261799072650750498679059581417775997642558862758595806102272', 'gens': [1, 2, 4, 6, 7, 8, 9], 'pres': [9, -2, -2, -2, -2, -5, -2, 2, 2, 2, 109, 46, 2595, 1308, 102, 2884, 1453, 21605, 10814, 6512, 1391, 50406, 25215, 663, 3822, 4354, 2563, 2465, 5309]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1085736031, 3876913498, 1089034753, 1087243448, 1087431672, 969133475, 1087374465, 1077385976, 1076435457]}, 'Perm': {'d': 14, 'gens': [999094447, 7784985720, 12933044760, 12945463200, 14374039680, 1967535360, 14382387360, 14382387365, 16]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^6:D_{10}', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}
