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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '432.362', 'ambient_counter': 362, 'ambient_order': 432, 'ambient_tex': 'D_{36}:C_6', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 4, 'counter': 77, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '432.362.18.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '18.d1.a1', 'outer_equivalence': False, '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': 18, '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': '24.10', 'subgroup_hash': 10, 'subgroup_order': 24, 'subgroup_tex': 'C_3\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '432.362', 'aut_centralizer_order': 12, 'aut_label': '18.d1', 'aut_quo_index': None, 'aut_stab_index': 18, 'aut_weyl_group': '4.2', 'aut_weyl_index': 216, 'centralizer': '36.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.h1.a1', '9.a1.a1'], 'contains': ['36.c1.a1', '36.f1.b1', '36.j1.a1', '54.c1.a1'], 'core': '108.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [1695, -1, 9412, -1, 8269, -1, 9620, -1], 'generators': [1, 162, 148, 216], 'label': '432.362.18.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '18.d1.a1', 'projective_image': '216.111', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.d1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2], 'aut_gens': [[1, 2], [7, 14], [13, 22], [1, 10], [13, 2]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [24, 126, 1, 414], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [3, 1, 2, 1], [4, 2, 1, 1], [6, 1, 2, 1], [6, 2, 4, 1], [12, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [3, 1, 2], [4, 2, 1], [6, 1, 2], [6, 2, 4], [12, 2, 2]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [3, 1, 2, 1], [4, 2, 1, 1], [6, 1, 2, 1], [6, 2, 2, 2], [12, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '12.5', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 14], [13, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 12], [2, 3]], 'label': '24.10', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 5, 'linQ_dim': 4, 'linQ_dim_count': 5, 'linR_count': 5, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 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': 8, 'number_conjugacy_classes': 15, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 20, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 5], [3, 2], [4, 2], [6, 10], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 2], 'outer_gens': [[1, 10], [7, 2]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 1]], 'representations': {'PC': {'code': 19095777, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -3, 113, 21, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16563488, 35813297]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 2064, 1717]}, 'Perm': {'d': 7, 'gens': [1680, 24, 4, 744]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times D_4', 'transitive_degree': 12, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 36, 'aut_gen_orders': [2, 2, 6, 36], 'aut_gens': [[1, 2, 12], [1, 218, 12], [1, 2, 204], [1, 2, 84], [109, 242, 12]], 'aut_group': '864.2583', 'aut_hash': 2583, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 22, 'aut_perms': [33350983693575167563, 50651938115792684319, 43805403314537441507, 945314625007338996230], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [2, 9, 2, 1], [2, 18, 2, 1], [3, 2, 1, 1], [3, 3, 1, 2], [4, 2, 1, 1], [4, 18, 1, 1], [6, 2, 1, 1], [6, 3, 1, 2], [6, 4, 2, 1], [6, 6, 2, 2], [6, 9, 2, 2], [6, 18, 2, 2], [9, 6, 1, 3], [12, 4, 1, 1], [12, 6, 1, 2], [12, 18, 1, 2], [18, 6, 1, 3], [18, 12, 2, 3], [36, 12, 1, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_{36}:C_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '108.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 108, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{18}:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [2, 9, 2], [2, 18, 2], [3, 2, 1], [3, 3, 2], [4, 2, 1], [4, 18, 1], [6, 2, 1], [6, 3, 2], [6, 4, 2], [6, 6, 4], [6, 9, 4], [6, 18, 4], [9, 6, 3], [12, 4, 1], [12, 6, 2], [12, 18, 2], [18, 6, 3], [18, 12, 6], [36, 12, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '216.111', 'commutator_count': 1, 'commutator_label': '18.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 362, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['54.6', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [2, 9, 1, 2], [2, 18, 1, 2], [3, 2, 1, 1], [3, 3, 2, 1], [4, 2, 1, 1], [4, 18, 1, 1], [6, 2, 1, 1], [6, 3, 2, 1], [6, 4, 1, 2], [6, 6, 2, 2], [6, 9, 2, 2], [6, 18, 2, 2], [9, 6, 1, 1], [9, 6, 2, 1], [12, 4, 1, 1], [12, 6, 2, 1], [12, 18, 2, 1], [18, 6, 1, 1], [18, 6, 2, 1], [18, 12, 1, 2], [18, 12, 2, 2], [36, 12, 1, 1], [36, 12, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 26208, 'exponent': 36, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 1]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '72.48', 'hash': 362, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 18], 'inner_gens': [[1, 2, 228], [1, 2, 348], [217, 98, 12]], 'inner_hash': 111, 'inner_nilpotent': False, 'inner_order': 216, 'inner_split': True, 'inner_tex': 'D_{18}:C_6', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 24], [2, 18], [4, 3], [6, 4], [12, 1]], 'label': '432.362', 'linC_count': 24, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D36:C6', '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': 37, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 50, 'number_divisions': 35, 'number_normal_subgroups': 56, 'number_subgroup_autclasses': 108, 'number_subgroup_classes': 170, 'number_subgroups': 750, 'old_label': None, 'order': 432, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 59], [3, 8], [4, 20], [6, 148], [9, 18], [12, 52], [18, 90], [36, 36]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [324, 0], 'outer_gens': [[109, 2, 12], [1, 218, 12]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 7], [6, 4], [8, 1], [12, 1]], 'representations': {'PC': {'code': 1687910333679833783225073984765269417235, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -3, -2, -2, -3, -3, 36, 6387, 4882, 1109, 80, 4631, 2748, 102, 2028, 2035, 166, 7069]}, 'GLZN': {'d': 2, 'p': 36, 'gens': [1089247, 385262, 1318384, 46681, 70633, 47089, 638377]}, 'Perm': {'d': 13, 'gens': [94434607, 16, 1, 124282920, 7, 649602000, 1093891800]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{36}:C_6', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
