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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '3888.jh', 'ambient_counter': 242, 'ambient_order': 3888, 'ambient_tex': 'C_3^3:S_3\\times S_4', 'central': False, 'central_factor': False, 'centralizer_order': 108, 'characteristic': False, 'core_order': 4, 'counter': 453, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3888.jh.108.bn1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '108.bn1.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': 108, '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': '36.14', 'subgroup_hash': 14, 'subgroup_order': 36, 'subgroup_tex': 'C_6^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3888.jh', 'aut_centralizer_order': 108, 'aut_label': '108.bn1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '24.14', 'aut_weyl_index': 972, 'centralizer': '36.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['36.a1.a1', '36.bf1.a1', '36.bg1.a1', '54.z1.a1', '54.bh1.a1', '54.bp1.a1'], 'contains': ['216.e1.a1', '324.ba1.a1', '324.bb1.a1', '324.bb1.c1'], 'core': '972.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [5493, -1, 5377, -1, 6450, -1, 7095, -1], 'generators': [23, 14883120, 1440644520, 7], 'label': '3888.jh.108.bn1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '36.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '108.bn1.a1', 'projective_image': '3888.jh', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '108.bn1.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '36.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 6, 4, 12], 'aut_gens': [[1, 6], [1, 34], [8, 5], [25, 34], [34, 35]], 'aut_group': '288.851', 'aut_hash': 851, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 11, 'aut_perms': [131784, 3762265, 24173544, 16148907], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 8, 1], [6, 1, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times \\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '288.851', 'autcent_hash': 851, 'autcent_nilpotent': False, 'autcent_order': 288, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_3\\times \\GL(2,3)', '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], [3, 1, 8], [6, 1, 24]], 'center_label': '36.14', 'center_order': 36, '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', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 4], [6, 1, 2, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.14', 'hash': 14, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 6], [1, 6]], '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, 36]], 'label': '36.14', 'linC_count': 144, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 36, 'linQ_dim': 4, 'linQ_dim_count': 36, 'linR_count': 36, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6^2', 'ngens': 4, 'nilpotency_class': 1, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 36, 'number_divisions': 20, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 30, 'number_subgroups': 30, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8], [6, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 6, 4, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 34], [8, 5], [25, 34], [34, 35]], 'outer_group': '288.851', 'outer_hash': 851, 'outer_nilpotent': False, 'outer_order': 288, 'outer_permdeg': 11, 'outer_perms': [131784, 3762265, 24173544, 16148907], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_3\\times \\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 16]], 'representations': {'PC': {'code': 344149, 'gens': [1, 3], 'pres': [4, -2, -3, -2, -3, 8, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26188475, 35931481]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1718, 1030]}, 'Perm': {'d': 10, 'gens': [362880, 5040, 240, 4]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2', 'transitive_degree': 36, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [12, 18, 6, 6], 'aut_gens': [[90852489, 573365525, 1086105013], [1086109933, 7716261, 497675658], [2115963370, 14883122, 2683665378], [101698577, 573355446, 2961841809], [3561840129, 14888285, 1495232773]], 'aut_group': None, 'aut_hash': 6340318746942157276, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 33, 'aut_perms': [5654596573826375708860222273914235125, 8023912229078664128328241938772186892, 813008164640688393860599339578781969, 5905798883844225990632227033684846724], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 3, 1], [3, 8, 1, 1], [3, 16, 1, 1], [3, 18, 1, 1], [3, 24, 2, 1], [3, 48, 3, 1], [3, 144, 1, 1], [4, 6, 1, 1], [4, 162, 1, 1], [6, 6, 1, 1], [6, 9, 2, 1], [6, 12, 1, 1], [6, 18, 2, 1], [6, 18, 3, 1], [6, 27, 2, 1], [6, 36, 3, 1], [6, 54, 1, 1], [6, 81, 2, 1], [6, 108, 1, 1], [6, 162, 2, 1], [6, 216, 1, 1], [6, 216, 2, 1], [9, 18, 2, 1], [9, 144, 2, 1], [12, 12, 1, 1], [12, 18, 2, 1], [12, 36, 3, 1], [12, 108, 1, 1], [12, 162, 2, 1], [18, 54, 2, 1], [18, 108, 2, 1], [36, 108, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.C_3^4.C_2^3', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 6340318746942157276, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.C_3^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 6, 1], [2, 27, 1], [2, 81, 1], [2, 162, 1], [3, 2, 1], [3, 3, 2], [3, 6, 3], [3, 8, 1], [3, 16, 1], [3, 18, 1], [3, 24, 2], [3, 48, 3], [3, 144, 1], [4, 6, 1], [4, 162, 1], [6, 6, 1], [6, 9, 2], [6, 12, 1], [6, 18, 5], [6, 27, 2], [6, 36, 3], [6, 54, 1], [6, 81, 2], [6, 108, 1], [6, 162, 2], [6, 216, 3], [9, 18, 2], [9, 144, 2], [12, 12, 1], [12, 18, 2], [12, 36, 3], [12, 108, 1], [12, 162, 2], [18, 54, 2], [18, 108, 2], [36, 108, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '3888.jh', 'commutator_count': 1, 'commutator_label': '972.525', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 242, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['162.19', 1], ['24.12', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 3], [3, 8, 1, 1], [3, 16, 1, 1], [3, 18, 1, 1], [3, 24, 2, 1], [3, 48, 1, 3], [3, 144, 1, 1], [4, 6, 1, 1], [4, 162, 1, 1], [6, 6, 1, 1], [6, 9, 2, 1], [6, 12, 1, 1], [6, 18, 1, 3], [6, 18, 2, 1], [6, 27, 2, 1], [6, 36, 1, 3], [6, 54, 1, 1], [6, 81, 2, 1], [6, 108, 1, 1], [6, 162, 2, 1], [6, 216, 1, 1], [6, 216, 2, 1], [9, 18, 1, 2], [9, 144, 1, 2], [12, 12, 1, 1], [12, 18, 2, 1], [12, 36, 1, 3], [12, 108, 1, 1], [12, 162, 2, 1], [18, 54, 1, 2], [18, 108, 1, 2], [36, 108, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 816480, 'exponent': 36, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 1, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '432.746', 'hash': 1865359641169939977, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 6, 12], 'inner_gens': [[90852489, 1451535134, 2690842337], [1487940498, 573365525, 2119632497], [4120307302, 1444353246, 1086105013]], 'inner_hash': 1865359641169939977, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'C_3^3:S_3\\times S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 4], [2, 10], [3, 12], [4, 4], [6, 18], [9, 8], [12, 3], [18, 6]], 'label': '3888.jh', 'linC_count': 24, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 24, 'linQ_dim': 9, 'linQ_dim_count': 24, 'linR_count': 24, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3:S3*S4', 'ngens': 3, '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, 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': 40, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 65, 'number_divisions': 55, 'number_normal_subgroups': 37, 'number_subgroup_autclasses': 556, 'number_subgroup_classes': 814, 'number_subgroups': 20184, 'old_label': None, 'order': 3888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 279], [3, 404], [4, 168], [6, 1584], [9, 324], [12, 588], [18, 324], [36, 216]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[2119632489, 7716245, 490383373]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [3, 4], [4, 4], [6, 18], [12, 5], [18, 10]], 'representations': {'PC': {'code': '659750185664738126159132420975247302750289369245548774734957524922304615990625716652758156588328555', 'gens': [1, 2, 4, 5, 7], 'pres': [9, 2, 2, 3, 3, 2, 3, 2, 3, 3, 30456, 37549, 46, 218, 74739, 444, 30784, 7456, 130, 73877, 154230, 53313, 46518, 11940, 4956, 1374, 186, 259207, 20752, 9097, 3499, 3076, 286, 209960]}, 'Perm': {'d': 13, 'gens': [90852489, 573365525, 1086105013]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3\\times S_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
