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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '648.616', 'ambient_counter': 616, 'ambient_order': 648, 'ambient_tex': 'C_2\\times A_4\\times \\He_3', 'central': False, 'central_factor': False, 'centralizer_order': 216, 'characteristic': False, 'core_order': 1, 'counter': 70, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '648.616.324.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '324.b1', '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': 324, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '648.616', 'aut_centralizer_order': 10368, 'aut_label': '324.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '1.1', 'aut_weyl_index': 31104, 'centralizer': '3.c1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['108.e1', '108.f1', '162.a1', '162.b1', '162.c1'], 'contains': ['648.a1'], 'core': '648.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [5006, -1, 6163, -1], 'generators': [324], 'label': '648.616.324.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '162.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.c1', 'old_label': '324.b1', 'projective_image': '648.616', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '324.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', '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, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], '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, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '54.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [24, 6, 2], 'aut_gens': [[1, 3, 18, 108], [73, 177, 595, 540], [252, 597, 542, 594], [433, 219, 523, 540]], 'aut_group': None, 'aut_hash': 3163708608538022825, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 31104, 'aut_permdeg': 50, 'aut_perms': [13927035929402594995215764221303118860237194644908343987210989874, 17063480245275760516167432037756468070425047509715642300379707955, 24344200467973091093393197550422008849519285467787911024678803413], 'aut_phi_ratio': 144.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 3, 8, 1], [3, 4, 6, 1], [3, 12, 16, 1], [6, 1, 2, 1], [6, 3, 2, 2], [6, 3, 8, 1], [6, 4, 6, 1], [6, 9, 8, 2], [6, 12, 16, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSU(3,2).C_6^2.D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '27.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 27, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '1152.157461', 'autcentquo_hash': 7371672288863537087, 'autcentquo_nilpotent': False, 'autcentquo_order': 1152, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times \\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 1, 2], [3, 3, 8], [3, 4, 6], [3, 12, 16], [6, 1, 2], [6, 3, 12], [6, 4, 6], [6, 9, 16], [6, 12, 16]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '108.41', 'commutator_count': 1, 'commutator_label': '12.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 616, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 1], ['27.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 3, 2, 4], [3, 4, 2, 3], [3, 12, 2, 8], [6, 1, 2, 1], [6, 3, 2, 6], [6, 4, 2, 3], [6, 9, 2, 8], [6, 12, 2, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4095, 'exponent': 6, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[9, 0, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '216.173', 'hash': 616, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 6, 2], 'inner_gens': [[1, 3, 234, 108], [1, 3, 396, 162], [433, 381, 18, 108], [1, 57, 18, 108]], 'inner_hash': 41, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': True, 'inner_tex': 'C_3^2\\times A_4', 'inner_used': [1, 2, 3], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 54], [3, 30], [9, 4]], 'label': '648.616', '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*A4*He3', 'ngens': 7, '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 88, 'number_divisions': 46, 'number_normal_subgroups': 78, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 268, 'number_subgroups': 1154, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 242], [6, 398]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 6, 4, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[2, 15, 18, 594], [433, 15, 20, 162], [288, 3, 56, 108], [289, 435, 523, 108]], 'outer_group': '288.851', 'outer_hash': 851, 'outer_nilpotent': False, 'outer_order': 288, 'outer_permdeg': 11, 'outer_perms': [367944, 1257385, 13119240, 21874347], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_3\\times \\GL(2,3)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 26], [3, 2], [6, 14], [18, 2]], 'representations': {'PC': {'code': 14958035244388889146904175094895957, 'gens': [1, 2, 4, 6], 'pres': [7, 3, 2, 3, 2, 3, 2, 3, 36, 6555, 3706, 1613, 80, 16384, 2280, 3421, 124]}, 'GLZN': {'d': 2, 'p': 36, 'gens': [47089, 1622293, 191563, 909811, 1166401, 47305, 70633]}, 'Perm': {'d': 15, 'gens': [127, 107416915504, 201312402480, 292724449920, 351307756800, 7, 126]}}, 'schur_multiplier': [3, 3, 3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times A_4\\times \\He_3', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
