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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1200.481', 'ambient_counter': 481, 'ambient_order': 1200, 'ambient_tex': '\\SL(2,5):D_5', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 2, 'counter': 29, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1200.481.50.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '50.b1.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': 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': '24.3', 'subgroup_hash': 3, 'subgroup_order': 24, 'subgroup_tex': '\\SL(2,3)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1200.481', 'aut_centralizer_order': 40, 'aut_label': '50.b1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '24.12', 'aut_weyl_index': 200, 'centralizer': '60.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['10.a1.a1', '10.c1.a1', '25.a1.a1'], 'contains': ['150.c1.a1', '200.a1.a1'], 'core': '600.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [9818, -1, 9841, -1, 9816, -1, 9834, -1], 'generators': [122109387504, 49052833033, 38285733465, 119692785530], 'label': '1200.481.50.b1.a1', 'mobius_quo': None, 'mobius_sub': -5, 'normal_closure': '10.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '50.b1.a1', 'projective_image': '600.146', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '50.b1.a1', 'subgroup_fusion': None, 'weyl_group': '12.3'}
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gps_groups • Show schema
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{'Agroup': False, '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': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[31, 21], [34, 67], [31, 53], [46, 21], [75, 15]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 4, 2, 1], [4, 6, 1, 1], [6, 4, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4', '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': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 4, 2], [4, 6, 1], [6, 4, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '8.4', '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': 3, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 4, 2, 1], [4, 6, 1, 1], [6, 4, 2, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 16, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, -1, 1], [2, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '12.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2], 'inner_gens': [[31, 79], [46, 21]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 3], [2, 3], [3, 1]], 'label': '24.3', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,3)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, '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': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 7, 'number_divisions': 5, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 7, 'number_subgroup_classes': 7, 'number_subgroups': 15, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 6], [6, 8]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [28], 'outer_gens': [[34, 41]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 2], [3, 1], [4, 1]], 'representations': {'PC': {'code': 3312657896, 'gens': [1, 2, 3], 'pres': [4, -3, -2, 2, -2, 145, 37, 110, 78, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [27313555, 20279859]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [31, 21], 'family': 'SL'}, {'d': 2, 'q': 3, 'gens': [362, 766], 'family': 'SU'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 37]}, 'Perm': {'d': 8, 'gens': [576, 28776, 16577, 5167]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,3)', 'transitive_degree': 8, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [4, 30, 4, 10, 2, 2], 'aut_gens': [[122109387504, 73876491433, 149770320628, 135774457122, 69679291610], [122109387504, 97993917916, 149770320628, 43907530549, 45561477502], [122109387504, 73876491433, 119692785530, 1574781090, 149660218931], [122109387504, 25631485987, 4706425968, 9398928656, 95027186848], [122109387504, 73876491433, 149770320628, 20931908132, 65141270182], [122109387504, 73876491433, 149770320628, 54716187408, 119776274790], [122109387504, 49748912470, 149770320628, 114216429080, 65141270182]], 'aut_group': '4800.cf', 'aut_hash': 6606067472031944380, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4800, 'aut_permdeg': 30, 'aut_perms': [212518488884132706665152006552464, 247362072801408138700048165150067, 246377761711248944963503173262685, 106417351784188702631950239984768, 44525005985431816212575533307088, 230202018455647724061594054475848], 'aut_phi_ratio': 15.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 150, 1, 1], [3, 20, 1, 1], [4, 5, 2, 1], [4, 30, 1, 1], [5, 2, 2, 1], [5, 12, 2, 1], [5, 24, 4, 1], [6, 20, 1, 1], [10, 2, 2, 1], [10, 12, 2, 1], [10, 24, 4, 1], [12, 100, 2, 1], [15, 40, 2, 1], [20, 60, 2, 1], [20, 60, 4, 1], [30, 40, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times F_5\\times S_5', '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': 60, 'autcentquo_group': '2400.db', 'autcentquo_hash': 3070016846447328869, 'autcentquo_nilpotent': False, 'autcentquo_order': 2400, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 150, 1], [3, 20, 1], [4, 5, 2], [4, 30, 1], [5, 2, 2], [5, 12, 2], [5, 24, 4], [6, 20, 1], [10, 2, 2], [10, 12, 2], [10, 24, 4], [12, 100, 2], [15, 40, 2], [20, 60, 6], [30, 40, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '600.146', 'commutator_count': 1, 'commutator_label': '600.54', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '60.5'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 481, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 150, 1, 1], [3, 20, 1, 1], [4, 5, 2, 1], [4, 30, 1, 1], [5, 2, 2, 1], [5, 12, 2, 1], [5, 24, 2, 2], [6, 20, 1, 1], [10, 2, 2, 1], [10, 12, 2, 1], [10, 24, 2, 2], [12, 100, 2, 1], [15, 40, 2, 1], [20, 60, 2, 1], [20, 60, 4, 1], [30, 40, 2, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 114, 'exponent': 60, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 1, 4], [8, 1, 2], [12, 1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '600.146', 'hash': 481, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [1, 5, 2, 6, 2], 'inner_gens': [[122109387504, 73876491433, 149770320628, 135774457122, 69679291610], [122109387504, 73876491433, 149770320628, 76469215445, 21444051669], [122109387504, 73876491433, 149770320628, 51333236783, 69679291610], [122109387504, 49748912470, 65040384091, 135774457122, 21444051669], [122109387504, 49748912470, 149770320628, 43907530549, 69679291610]], 'inner_hash': 146, 'inner_nilpotent': False, 'inner_order': 600, 'inner_split': False, 'inner_tex': 'D_5\\times A_5', 'inner_used': [2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 2], [2, 6], [3, 4], [4, 8], [5, 2], [6, 6], [8, 4], [10, 2], [12, 2]], 'label': '1200.481', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,5):D5', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 36, 'number_divisions': 20, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 70, 'number_subgroups': 1562, 'old_label': None, 'order': 1200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 151], [3, 20], [4, 40], [5, 124], [6, 20], [10, 124], [12, 200], [15, 80], [20, 360], [30, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [10030720628, 30527346876, 145124556878], 'outer_gens': [[122109387504, 73876491433, 105184217503, 57115357937, 69679291610], [122109387504, 73876491433, 149770320628, 54716187408, 119776274790], [122109387504, 97993917916, 149770320628, 135774457122, 149660218931]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 29, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 3], [5, 2], [6, 2], [8, 4], [12, 3], [16, 2], [20, 1], [24, 1]], 'representations': {'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 73876491433, 149770320628, 135774457122, 69679291610]}, 'Perm': {'d': 29, 'gens': [359476225705797099128504841682, 983230330386097398936787153225]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,5):D_5', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
