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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '648.572', 'ambient_counter': 572, 'ambient_order': 648, 'ambient_tex': 'C_3^3:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 108, 'characteristic': True, 'core_order': 36, 'counter': 24, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '648.572.18.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '18.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '18.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 18, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3:S_3', 'simple': False, 'solvable': True, 'special_labels': ['S'], 'split': False, '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': '648.572', 'aut_centralizer_order': 36, 'aut_label': '18.a1', 'aut_quo_index': 4, 'aut_stab_index': 1, 'aut_weyl_group': '72.46', 'aut_weyl_index': 36, 'centralizer': '6.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['6.a1', '6.b1', '9.a1'], 'contains': ['36.c1', '54.a1', '54.b1'], 'core': '18.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5767, 2399, 4935, 1452], 'generators': [378, 324, 2, 36], 'label': '648.572.18.a1', 'mobius_quo': -3, 'mobius_sub': -27, 'normal_closure': '18.a1', 'normal_contained_in': ['6.a1', '6.b1'], 'normal_contains': ['54.a1', '54.b1', '72.a1'], 'normalizer': '1.a1', 'old_label': '18.a1', 'projective_image': '72.43', 'quotient_action_image': '6.1', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '18.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 6, 6, 6], 'aut_gens': [[1, 6, 18, 108], [451, 24, 90, 540], [433, 228, 18, 558], [551, 186, 360, 558], [253, 150, 18, 108], [307, 420, 18, 108]], 'aut_group': '2592.jb', 'aut_hash': 6898096242996517164, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2592, 'aut_permdeg': 24, 'aut_perms': [69690952200231007002212, 76534936139814304535949, 160948946215292178707253, 13541764093129214228105, 10169710627794712883484], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 18, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 24, 3, 1], [3, 24, 6, 1], [4, 18, 1, 1], [6, 3, 2, 1], [6, 3, 6, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 6, 6, 1], [6, 18, 2, 1], [6, 18, 6, 1], [12, 18, 2, 1], [12, 18, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_6^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '432.523', 'autcentquo_hash': 523, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2:D_6', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 18, 1], [3, 1, 8], [3, 6, 3], [3, 24, 9], [4, 18, 1], [6, 3, 8], [6, 6, 9], [6, 18, 8], [12, 18, 8]], 'center_label': '9.2', 'center_order': 9, 'central_product': True, 'central_quotient': '72.43', 'commutator_count': 2, 'commutator_label': '108.22', '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': 572, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['216.95', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 18, 1, 1], [3, 1, 2, 4], [3, 6, 1, 1], [3, 6, 2, 1], [3, 24, 1, 3], [3, 24, 2, 3], [4, 18, 1, 1], [6, 3, 2, 4], [6, 6, 1, 1], [6, 6, 2, 4], [6, 18, 2, 4], [12, 18, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 49140, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '216.164', 'hash': 572, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 2, 6], 'inner_gens': [[1, 12, 18, 558], [13, 6, 342, 450], [1, 330, 18, 108], [307, 420, 18, 108]], 'inner_hash': 43, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'C_3:S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 12], [3, 30], [6, 9]], 'label': '648.572', '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': 'C3^3:S4', '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': 19, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 57, 'number_divisions': 33, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 196, 'number_subgroups': 1160, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 21], [3, 242], [4, 18], [6, 222], [12, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 3, 3], 'outer_gen_pows': [0, 0, 3, 0], 'outer_gens': [[5, 6, 18, 108], [5, 12, 90, 198], [1, 300, 18, 558], [37, 6, 18, 108]], 'outer_group': '36.10', 'outer_hash': 10, 'outer_nilpotent': False, 'outer_order': 36, 'outer_permdeg': 6, 'outer_perms': [316, 341, 378, 373], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [3, 2], [4, 4], [6, 15], [12, 4]], 'representations': {'PC': {'code': 3222718250755765020368323440170165, 'gens': [1, 3, 4, 6], 'pres': [7, 2, 3, 3, 2, 3, 2, 3, 14, 254, 1613, 80, 23441, 3169, 124, 22938, 2078]}, 'Perm': {'d': 16, 'gens': [87178293556, 43597501, 93405315013, 98356, 123858, 1313901388800, 2789705318400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], '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_4', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 3, 3], 'aut_gens': [[25, 3, 144], [25, 3, 243], [25, 147, 144], [122, 3, 144]], 'aut_group': '432.734', 'aut_hash': 734, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 432, 'aut_permdeg': 9, 'aut_perms': [42648, 223305, 287216], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2:\\GL(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': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 7, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '18.4', 'hash': 4, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[25, 4, 240], [29, 3, 144], [265, 3, 144]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 4]], 'label': '18.4', 'linC_count': 6, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 6, 'linQ_dim': 4, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3:S3', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 12, 'number_subgroups': 28, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 9], [3, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 0, 25, 25], 'outer_gens': [[25, 3, 243], [25, 240, 243], [25, 243, 244], [25, 240, 3]], 'outer_group': '24.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 4, 'outer_perms': [2, 4, 16, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 4]], 'representations': {'PC': {'code': 5475, 'gens': [1, 2, 3], 'pres': [3, -2, -3, -3, 25, 110]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26483560, 35931151, 16858733]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [11017, 8101, 13933]}, 'Perm': {'d': 6, 'gens': [25, 3, 144]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:S_3', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
