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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '6750.b', 'ambient_counter': 2, 'ambient_order': 6750, 'ambient_tex': '(C_5\\times C_{15}^2):C_6', 'central': False, 'central_factor': False, 'centralizer_order': 15, 'characteristic': False, 'core_order': 45, 'counter': 32, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6750.b.50.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '50.a1', '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': 50, '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': '135.3', 'subgroup_hash': 3, 'subgroup_order': 135, 'subgroup_tex': 'C_5\\times \\He_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6750.b', 'aut_centralizer_order': None, 'aut_label': '50.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '450.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1', '25.a1'], 'contains': ['150.a1', '150.d1', '250.a1'], 'core': '150.a1', 'coset_action_label': None, 'count': 25, 'diagramx': [9036, -1, 2223, -1], 'generators': [2, 1626, 300, 4500], 'label': '6750.b.50.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '25.a1', 'old_label': '50.a1', 'projective_image': '2250.g', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '50.a1', 'subgroup_fusion': None, 'weyl_group': '18.5'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '45.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 3, 4, 4, 2, 3, 3], 'aut_gens': [[1, 3, 9], [46, 52, 99], [1, 3, 63], [1, 3, 36], [1, 5, 9], [97, 98, 9], [51, 91, 9], [92, 6, 9], [91, 93, 9], [1, 93, 9]], 'aut_group': '1728.46191', 'aut_hash': 46191, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1728, 'aut_permdeg': 13, 'aut_perms': [284302920, 9, 16, 83841120, 261641520, 51700320, 176158200, 3933792000, 1092687120], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 8, 1], [5, 1, 4, 1], [15, 1, 8, 1], [15, 3, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times C_3^2:\\GL(2,3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '36.8', 'autcent_hash': 8, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3\\times C_{12}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '48.29', 'autcentquo_hash': 29, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(2,3)', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 8], [5, 1, 4], [15, 1, 8], [15, 3, 32]], 'center_label': '15.1', 'center_order': 15, 'central_product': True, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['27.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 4], [5, 1, 4, 1], [15, 1, 8, 1], [15, 3, 8, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 6, 'exponent': 15, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3, 5], 'faithful_reps': [[3, 0, 8]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '45.2', 'hash': 3, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 1], 'inner_gens': [[1, 93, 9], [46, 3, 9], [1, 3, 9]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': True, 'inner_tex': 'C_3^2', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 6, 'irrep_stats': [[1, 45], [3, 10]], 'label': '135.3', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*He3', '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 55, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 22, 'number_subgroups': 38, 'old_label': None, 'order': 135, 'order_factorization_type': 31, 'order_stats': [[1, 1], [3, 26], [5, 4], [15, 104]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 12, 4, 4], 'outer_gen_pows': [98, 0, 0, 0], 'outer_gens': [[92, 5, 99], [93, 95, 63], [52, 8, 9], [53, 5, 9]], 'outer_group': '192.951', 'outer_hash': 951, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 12, 'outer_perms': [18594007, 14893217, 258750120, 212422440], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [3, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [4, 1], [6, 1], [8, 4], [24, 1]], 'representations': {'PC': {'code': 141691991, 'gens': [1, 2, 3], 'pres': [4, -3, -3, -3, -5, 745, 46]}, 'GLZN': {'d': 2, 'p': 45, 'gens': [1458001, 152806, 91801, 91531]}, 'Perm': {'d': 14, 'gens': [20253915360, 40475635200, 96, 12461309280]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 15], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times \\He_3', 'transitive_degree': 45, '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': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [24, 24, 24, 12, 8], 'aut_gens': [[1, 6, 30, 450], [5893, 1374, 78, 1164], [2131, 2886, 2826, 3234], [6251, 282, 1422, 4908], [5401, 4050, 144, 1932], [913, 366, 5550, 5322]], 'aut_group': None, 'aut_hash': 1525260094185992089, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1296000, 'aut_permdeg': 575, 'aut_perms': 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'aut_phi_ratio': 720.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 125, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 75, 6, 1], [5, 2, 2, 1], [5, 6, 4, 1], [5, 6, 16, 1], [6, 125, 2, 1], [6, 375, 2, 1], [6, 375, 6, 1], [15, 2, 4, 1], [15, 6, 4, 1], [15, 6, 8, 1], [15, 6, 24, 1], [15, 6, 32, 1], [15, 6, 96, 1], [15, 150, 12, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_5\\times C_{15}).C_{15}.C_{12}^2.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 4315712504607614885, 'autcentquo_nilpotent': False, 'autcentquo_order': 144000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^3.C_6^2.C_2^4.C_2', 'cc_stats': [[1, 1, 1], [2, 125, 1], [3, 1, 2], [3, 3, 2], [3, 75, 6], [5, 2, 2], [5, 6, 20], [6, 125, 2], [6, 375, 8], [15, 2, 4], [15, 6, 164], [15, 150, 12]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '2250.g', 'commutator_count': 1, 'commutator_label': '375.7', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 125, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 75, 2, 3], [5, 2, 2, 1], [5, 6, 2, 10], [6, 125, 2, 1], [6, 375, 2, 4], [15, 2, 4, 1], [15, 6, 4, 41], [15, 150, 4, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 30, 'exponents_of_order': [3, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 96]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '2250.g', 'hash': 4747043094101760884, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [6, 5, 5, 15], 'inner_gens': [[1, 2706, 216, 3198], [4051, 6, 30, 450], [295, 6, 30, 450], [4483, 6, 30, 450]], 'inner_hash': 3638640786514561450, 'inner_nilpotent': False, 'inner_order': 2250, 'inner_split': False, 'inner_tex': 'C_3\\times C_5^3:C_6', 'inner_used': [1, 2, 4], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 18], [2, 18], [3, 4], [6, 184]], 'label': '6750.b', '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': '(C5*C15^2):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, 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': 18, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 224, 'number_divisions': 68, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 88, 'number_subgroup_classes': 328, 'number_subgroups': 9424, 'old_label': None, 'order': 6750, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 125], [3, 458], [5, 124], [6, 3250], [15, 2792]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 24, 6], 'outer_gen_pows': [0, 3, 0, 3], 'outer_gens': [[5, 5676, 2736, 1098], [4651, 18, 240, 3600], [151, 1446, 1668, 1992], [2251, 12, 210, 3150]], 'outer_group': '576.6547', 'outer_hash': 6547, 'outer_nilpotent': False, 'outer_order': 576, 'outer_permdeg': 16, 'outer_perms': [1313941668600, 1327353350400, 5493231083814, 5579411004847], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4.D_6^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [4, 1], [6, 2], [8, 4], [12, 10], [24, 41]], 'representations': {'PC': {'code': '1200683739143646001957950077860879643308354022076777021196151', 'gens': [1, 3, 4, 6], 'pres': [7, 2, 3, 5, 3, 5, 3, 5, 14, 56828, 16326, 6051, 39742, 108, 6934, 30356, 134321, 74982, 166, 137892, 74983]}, 'Perm': {'d': 24, 'gens': [55239425330337030235024, 28260974960271340365240]}}, 'schur_multiplier': [3, 15], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_5\\times C_{15}^2):C_6', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
