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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '43200.be', 'ambient_counter': 31, 'ambient_order': 43200, 'ambient_tex': 'A_4\\times A_5^2', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 1, 'counter': 118, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '43200.be.300.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '300.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': 300, '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': '144.190', 'subgroup_hash': 190, 'subgroup_order': 144, 'subgroup_tex': 'A_4\\times D_6', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '43200.be', 'aut_centralizer_order': None, 'aut_label': '300.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '10800.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['30.a1', '60.c1', '150.e1'], 'contains': ['600.a1', '600.m1', '600.q1', '900.d1', '900.f1', '1200.a1'], 'core': '43200.a1', 'coset_action_label': None, 'count': 300, 'diagramx': [6552, -1, 5528, -1], 'generators': [482721120, 131760, 81956902472, 482630400, 10080, 81474226719], 'label': '43200.be.300.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '150.e1', 'old_label': '300.b1', 'projective_image': '43200.be', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '300.b1', 'subgroup_fusion': None, 'weyl_group': '72.44'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 3, 3, 2, 2], 'aut_gens': [[5040, 120, 4, 7, 16, 45360], [5040, 120, 4, 7, 16, 80640], [5040, 120, 3, 7, 23, 45360], [5160, 120, 4, 7, 16, 45360], [5040, 120, 4, 16, 23, 45360], [85680, 120, 4, 7, 16, 45360], [5040, 120, 19, 7, 16, 45360], [5040, 120, 8, 7, 16, 45360]], 'aut_group': '288.1028', 'aut_hash': 1028, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 9, 'aut_perms': [7, 5040, 1, 5760, 30, 41040, 90720], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 9, 2, 1], [3, 2, 1, 1], [3, 4, 2, 1], [3, 8, 2, 1], [6, 2, 1, 1], [6, 4, 2, 1], [6, 6, 1, 2], [6, 8, 2, 1], [6, 12, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '144.183', 'autcentquo_hash': 183, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [3, 2, 1], [3, 4, 2], [3, 8, 2], [6, 2, 1], [6, 4, 2], [6, 6, 2], [6, 8, 2], [6, 12, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '72.44', 'commutator_count': 1, 'commutator_label': '12.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 190, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [3, 2, 1, 1], [3, 4, 2, 1], [3, 8, 2, 1], [6, 2, 1, 1], [6, 4, 2, 1], [6, 6, 1, 2], [6, 8, 2, 1], [6, 12, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '144.190', 'hash': 190, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 3, 2, 2, 3], 'inner_gens': [[5040, 120, 4, 7, 16, 80640], [5040, 120, 4, 7, 16, 45360], [5040, 120, 4, 23, 7, 45360], [5040, 120, 15, 7, 16, 45360], [5040, 120, 19, 7, 16, 45360], [85680, 120, 4, 7, 16, 45360]], 'inner_hash': 44, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'S_3\\times A_4', 'inner_used': [1, 3, 4, 6], 'irrC_degree': 6, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 12], [2, 6], [3, 4], [6, 2]], 'label': '144.190', 'linC_count': 18, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 6, 'linQ_dim': 5, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'A4*D6', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 15, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 24, 'number_divisions': 18, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 62, 'number_subgroup_classes': 82, 'number_subgroups': 336, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 31], [3, 26], [6, 86]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5160, 120, 4, 7, 16, 45360], [5040, 120, 3, 23, 16, 45360]], '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': 4, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [3, 4], [4, 2], [6, 2]], 'representations': {'PC': {'code': 6000145012420576674953, 'gens': [1, 2, 4, 5], 'pres': [6, -2, -2, -3, -2, 2, -3, 31, 873, 519, 1990, 826, 88, 1739]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [142763919005, 799054937903]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [12967, 18685, 2665, 1777, 1733, 1801]}, 'Perm': {'d': 9, 'gens': [5040, 120, 4, 7, 16, 45360]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4\\times D_6', 'transitive_degree': 18, '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': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [24, 12, 30], 'aut_gens': [[1037836800, 482630400, 1476921600, 69100064920, 70058159436], [482630400, 1520467200, 562464000, 76248885869, 482762167], [1037836800, 1520467200, 79833600, 82432326396, 82512241479], [1520467200, 1037836800, 43545600, 82914992916, 76767684646]], 'aut_group': '691200.g', 'aut_hash': 1857085433632322154, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 691200, 'aut_permdeg': 38, 'aut_perms': [200678443984620173108869747517205438390490998, 429400544971639395390776951565043186600075478, 374131159695911618719772098906153015431534742], 'aut_phi_ratio': 60.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 15, 2, 1], [2, 45, 2, 1], [2, 225, 1, 1], [2, 675, 1, 1], [3, 4, 2, 1], [3, 20, 2, 1], [3, 80, 4, 1], [3, 400, 1, 1], [3, 1600, 2, 1], [5, 12, 4, 1], [5, 144, 4, 1], [6, 60, 2, 1], [6, 60, 4, 1], [6, 300, 2, 1], [6, 900, 2, 2], [6, 1200, 1, 1], [6, 1200, 4, 1], [10, 36, 4, 1], [10, 180, 4, 1], [10, 432, 4, 1], [10, 540, 4, 1], [15, 48, 8, 1], [15, 240, 4, 1], [15, 576, 8, 1], [15, 960, 8, 1], [30, 720, 4, 1], [30, 720, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times S_5\\wr C_2', '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': 120, 'autcentquo_group': '691200.g', 'autcentquo_hash': 1857085433632322154, 'autcentquo_nilpotent': False, 'autcentquo_order': 691200, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times S_5\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 15, 2], [2, 45, 2], [2, 225, 1], [2, 675, 1], [3, 4, 2], [3, 20, 2], [3, 80, 4], [3, 400, 1], [3, 1600, 2], [5, 12, 4], [5, 144, 4], [6, 60, 6], [6, 300, 2], [6, 900, 4], [6, 1200, 5], [10, 36, 4], [10, 180, 4], [10, 432, 4], [10, 540, 4], [15, 48, 8], [15, 240, 4], [15, 576, 8], [15, 960, 8], [30, 720, 12]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '43200.be', 'commutator_count': 1, 'commutator_label': '14400.cf', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '60.5', '60.5'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['60.5', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 15, 1, 2], [2, 45, 1, 2], [2, 225, 1, 1], [2, 675, 1, 1], [3, 4, 2, 1], [3, 20, 1, 2], [3, 80, 2, 2], [3, 400, 1, 1], [3, 1600, 2, 1], [5, 12, 2, 2], [5, 144, 2, 2], [6, 60, 1, 2], [6, 60, 2, 2], [6, 300, 1, 2], [6, 900, 1, 2], [6, 900, 2, 1], [6, 1200, 1, 1], [6, 1200, 2, 2], [10, 36, 2, 2], [10, 180, 2, 2], [10, 432, 2, 2], [10, 540, 2, 2], [15, 48, 4, 2], [15, 240, 2, 2], [15, 576, 4, 2], [15, 960, 4, 2], [30, 720, 2, 2], [30, 720, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 684, 'exponent': 30, 'exponents_of_order': [6, 3, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[27, 1, 4], [36, 1, 4], [45, 1, 4], [48, 1, 1], [60, 1, 2], [75, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '43200.be', 'hash': 4943564940395828318, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 2, 3, 15, 2], 'inner_gens': [[1037836800, 482630400, 958003200, 70460864920, 70058159436], [1037836800, 482630400, 43545600, 70021780120, 70058159436], [1520467200, 1037836800, 1476921600, 69539149720, 70540789836], [482630400, 1520467200, 958003200, 69100064920, 75729932194], [1037836800, 482630400, 958003200, 82914910836, 70058159436]], 'inner_hash': 4943564940395828318, 'inner_nilpotent': False, 'inner_order': 43200, 'inner_split': True, 'inner_tex': 'A_4\\times A_5^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 27, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 27, 'irrep_stats': [[1, 3], [3, 13], [4, 6], [5, 6], [9, 16], [12, 14], [15, 14], [16, 3], [20, 6], [25, 3], [27, 4], [36, 4], [45, 4], [48, 1], [60, 2], [75, 1]], 'label': '43200.be', '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': False, 'name': 'A4*A5^2', 'ngens': 5, '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, 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, 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': 30, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 100, 'number_divisions': 51, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 542, 'number_subgroup_classes': 1145, 'number_subgroups': 285056, 'old_label': None, 'order': 43200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1023], [3, 3968], [5, 624], [6, 10560], [10, 4752], [15, 13632], [30, 8640]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [41056, 81474243388, 69020311115, 69020311115], 'outer_gens': [[1037836800, 482630400, 1476921600, 81554112284, 70058189791], [1037836800, 482630400, 1476921600, 62873090074, 76285083754], [1520467200, 482630400, 1001548800, 69063777527, 82994789807], [1037836800, 482630400, 1476921600, 69100065527, 82512159407]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 415, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 2], [5, 2], [6, 2], [8, 2], [10, 2], [12, 4], [15, 2], [16, 1], [18, 4], [20, 2], [24, 2], [25, 1], [30, 2], [32, 1], [36, 2], [40, 2], [48, 3], [50, 1], [54, 2], [60, 4], [72, 2], [75, 1], [90, 2]], 'representations': {'Perm': {'d': 14, 'gens': [1037836800, 482630400, 1476921600, 69100064920, 70058159436]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4\\times A_5^2', 'transitive_degree': 100, 'wreath_data': None, 'wreath_product': False}
