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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '6144.bb', 'ambient_counter': 28, 'ambient_order': 6144, 'ambient_tex': 'C_4^4.C_{24}', 'central': False, 'central_factor': False, 'centralizer_order': 1536, 'characteristic': False, 'core_order': 16, 'counter': 933, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6144.bb.192.HH', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '192.hg1', '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': 192, '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': '32.16', 'subgroup_hash': 16, 'subgroup_order': 32, 'subgroup_tex': 'C_2\\times C_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6144.bb', 'aut_centralizer_order': None, 'aut_label': '192.HH', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4.H', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['64.Q', '96.DW', '96.EN', '96.GY', '96.IF', '96.IH'], 'contains': ['384.K', '384.HS'], 'core': '384.K', 'coset_action_label': None, 'count': 16, 'diagramx': None, 'generators': [641733, 49681], 'label': '6144.bb.192.HH', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '24.F', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.H', 'old_label': '192.hg1', 'projective_image': '384.18167', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '192.HH', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '32.16', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 2, 2, 4, 2], 'aut_gens': [[1, 2], [17, 15], [1, 3], [1, 30], [1, 10], [1, 14]], 'aut_group': '32.48', 'aut_hash': 48, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 10, 'aut_perms': [455286, 374406, 1270441, 859248, 1], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 2, 2], [8, 1, 4, 2], [16, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4:C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.48', 'autcent_hash': 48, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4:C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 1, 4], [8, 1, 8], [16, 1, 16]], 'center_label': '32.16', 'center_order': 32, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 16, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['16.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [8, 1, 4, 2], [16, 1, 8, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 16, 'exponents_of_order': [5], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '4.2', 'hash': 16, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32]], 'label': '32.16', 'linC_count': 192, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 4, 'linQ_dim': 9, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C16', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 32, 'number_divisions': 10, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 14, 'number_subgroups': 14, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4], [8, 8], [16, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 2, 4, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[17, 15], [1, 3], [1, 30], [1, 10], [1, 14]], 'outer_group': '32.48', 'outer_hash': 48, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [455286, 374406, 1270441, 859248, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4:C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 2, 'primary_abelian_invariants': [2, 16], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2], [8, 2]], 'representations': {'PC': {'code': 17891342, 'gens': [1, 2], 'pres': [5, -2, 2, -2, -2, -2, 26, 42, 58]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [19665, 14742]}, 'Perm': {'d': 18, 'gens': [20916435456000, 355687428096000, 9703614452976, 4097506710982, 1313941673647]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 16], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{16}', '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': '384.5396', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [8, 8, 12, 8, 4, 8, 6, 12], 'aut_gens': [[998829, 754463, 1032711, 500487, 1032735], [55022, 500247, 753695, 762639, 508431], [368314, 245791, 508679, 762127, 1032223], [353395, 819713, 582169, 303377, 492047], [317174, 1016583, 516615, 254751, 507935], [62710, 253967, 507927, 500487, 491551], [241202, 33033, 40985, 844561, 1016351], [620015, 581657, 557833, 303361, 1032223], [745141, 303105, 819457, 582425, 1016335]], 'aut_group': None, 'aut_hash': 9173326487122815049, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1572864, 'aut_permdeg': 288, 'aut_perms': 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[16, 4, 64], [16, 12, 64], [24, 16, 64], [48, 16, 128]], 'center_label': '64.83', 'center_order': 64, 'central_product': True, 'central_quotient': '96.72', 'commutator_count': 1, 'commutator_label': '16.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 28, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1], ['768.1083570', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 3, 1, 8], [3, 16, 2, 1], [4, 1, 2, 12], [4, 3, 2, 12], [4, 6, 1, 16], [4, 6, 2, 24], [6, 16, 2, 7], [8, 1, 4, 8], [8, 3, 4, 8], [8, 6, 4, 16], [12, 16, 4, 12], [16, 4, 8, 8], [16, 12, 8, 8], [24, 16, 8, 8], [48, 16, 16, 8]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 43680, 'exponent': 48, 'exponents_of_order': [11, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '64.246', 'frattini_quotient': '96.228', 'hash': 3133702695541003383, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 4, 4, 2, 2], 'inner_gens': [[998829, 1032727, 253983, 1041175, 492047], [744885, 754463, 1032711, 500487, 1032735], [990381, 754463, 1032711, 500487, 1032735], [1015725, 754463, 1032711, 500487, 1032735], [1015725, 754463, 1032711, 500487, 1032735]], 'inner_hash': 72, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': False, 'inner_tex': 'C_4^2:C_6', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 384], [3, 128], [6, 128]], 'label': '6144.bb', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 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8, 2, 4], 'outer_gen_pows': [49665, 32769, 146619, 32769, 32769, 32769, 933795, 378228], 'outer_gens': [[55022, 500247, 753695, 762639, 508431], [368314, 245791, 508679, 762127, 1032223], [353395, 819713, 582169, 303377, 492047], [317174, 1016583, 516615, 254751, 507935], [62710, 253967, 507927, 500487, 491551], [241202, 33033, 40985, 844561, 1016351], [620015, 581657, 557833, 303361, 1032223], [745141, 303105, 819457, 582425, 1016335]], 'outer_group': None, 'outer_hash': 793570606469686138, 'outer_nilpotent': True, 'outer_order': 16384, 'outer_permdeg': 256, 'outer_perms': 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'509873775914174022841184100850348418057930660757857287018008151124698500575189229469788339970638345840404177616022047779504533109656585288939049985897086475496517760', 'gens': [1, 6, 8, 10, 12], 'pres': [12, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 24, 61, 98, 135, 41477, 95057, 31133, 23801, 7829, 209, 32262, 84690, 8094, 21210, 2070, 133639, 76051, 26527, 19051, 6679, 283, 217736, 5204, 54464, 1340, 13664, 288009, 74901, 72033, 18765, 18057, 357, 677387, 255767, 169379, 63983, 42395]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [32777, 33025, 32785, 98307, 100557, 294921, 124284, 33281, 557073, 573953, 713485, 828161]}}, 'schur_multiplier': [2, 2, 4, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 48], 'solvability_type': 16, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_4^4.C_{24}', 'transitive_degree': 384, 'wreath_data': None, 'wreath_product': False}
