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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.9542', 'ambient_counter': 9542, 'ambient_order': 960, 'ambient_tex': 'C_{60}.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 60, 'counter': 119, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.9542.8.bd1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8.bd1.b1', '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': 8, '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': '120.14', 'subgroup_hash': 14, 'subgroup_order': 120, 'subgroup_tex': 'C_{15}:Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.9542', 'aut_centralizer_order': 8, 'aut_label': '8.bd1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '480.1197', 'aut_weyl_index': 32, 'centralizer': '480.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.n1.a1', '4.y1.b1', '4.bc1.a1'], 'contains': ['16.e1.b1', '16.j1.a1', '16.k1.b1', '24.bd1.b1', '40.bd1.b1'], 'core': '16.e1.b1', 'coset_action_label': None, 'count': 2, 'diagramx': [3031, -1, 6429, -1, 2151, -1, 7634, -1], 'generators': [517, 48, 32, 14, 192], 'label': '960.9542.8.bd1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.n1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.d1.a1', 'old_label': '8.bd1.b1', 'projective_image': '480.1123', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.bd1.b1', 'subgroup_fusion': None, 'weyl_group': '240.202'}
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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': 2, 'aut_exponent': 60, 'aut_gen_orders': [2, 2, 4, 30], 'aut_gens': [[1, 2, 24], [13, 2, 24], [1, 10, 24], [1, 10, 72], [17, 62, 24]], 'aut_group': '480.1197', 'aut_hash': 1197, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 480, 'aut_permdeg': 12, 'aut_perms': [90720, 3669840, 3719538, 79874711], 'aut_phi_ratio': 15.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 6, 1, 1], [4, 10, 1, 1], [4, 30, 1, 1], [5, 2, 2, 1], [6, 2, 1, 1], [10, 2, 2, 1], [12, 10, 2, 1], [15, 4, 2, 1], [20, 6, 4, 1], [30, 4, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '120.36', 'autcentquo_hash': 36, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 1], [4, 6, 1], [4, 10, 1], [4, 30, 1], [5, 2, 2], [6, 2, 1], [10, 2, 2], [12, 10, 2], [15, 4, 2], [20, 6, 4], [30, 4, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '60.8', 'commutator_count': 1, 'commutator_label': '30.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 6, 1, 1], [4, 10, 1, 1], [4, 30, 1, 1], [5, 2, 2, 1], [6, 2, 1, 1], [10, 2, 2, 1], [12, 10, 2, 1], [15, 4, 2, 1], [20, 6, 4, 1], [30, 4, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 60, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, -1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '60.8', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 6, 5], 'inner_gens': [[1, 22, 24], [5, 2, 96], [1, 50, 24]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 60, 'inner_split': True, 'inner_tex': 'S_3\\times D_5', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 13], [4, 4]], 'label': '120.14', 'linC_count': 26, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 7, 'linQ_dim': 10, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C15:Q8', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 21, 'number_divisions': 13, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 24, 'number_subgroups': 76, 'old_label': None, 'order': 120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 46], [5, 4], [6, 2], [10, 4], [12, 20], [15, 8], [20, 24], [30, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 10, 24], [1, 2, 48]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 3], [8, 3]], 'representations': {'PC': {'code': 152412364976812055559, 'gens': [1, 2, 5], 'pres': [5, -2, -2, -2, -3, -5, 60, 221, 26, 302, 42, 323, 1209]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [302101, 40966, 252261, 297011, 27181]}, 'Perm': {'d': 16, 'gens': [1508416318087, 2891341112400, 4302991486080, 5760, 37]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}:Q_8', 'transitive_degree': 120, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [4, 4, 4, 20, 4, 4, 4, 2], 'aut_gens': [[1, 2, 4, 96], [25, 562, 68, 912], [1, 50, 28, 336], [49, 506, 772, 672], [49, 554, 844, 144], [49, 50, 244, 720], [25, 562, 476, 288], [73, 530, 820, 672], [1, 2, 860, 912]], 'aut_group': None, 'aut_hash': 5847347909673561214, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15360, 'aut_permdeg': 36, 'aut_perms': [365347223059961355309501428406104385736271, 306736440428684102883410054746056272919500, 143916011655617118833605859357523320016248, 148418686559765272388409731928532605725749, 114070573906702359597662280509065716739310, 44585318717381938965858760039188030095684, 371680084161279150933014742425588249172345, 70641481942879119281479794469409038266023], 'aut_phi_ratio': 60.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 6, 2, 1], [2, 12, 2, 1], [2, 20, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 6, 2, 1], [4, 20, 1, 1], [4, 60, 2, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 8, 1, 1], [6, 40, 1, 1], [8, 10, 2, 1], [8, 20, 1, 1], [8, 60, 2, 1], [10, 2, 2, 1], [10, 4, 2, 1], [10, 4, 4, 1], [10, 12, 4, 1], [10, 12, 8, 1], [12, 2, 2, 1], [12, 4, 1, 1], [12, 8, 1, 1], [12, 40, 1, 1], [15, 4, 2, 1], [20, 4, 2, 2], [20, 4, 4, 1], [20, 12, 4, 1], [24, 20, 2, 2], [30, 4, 2, 1], [30, 8, 2, 1], [30, 8, 4, 1], [60, 4, 4, 1], [60, 8, 2, 1], [60, 8, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:(C_2^3.C_2^6.C_2)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '960.11377', 'autcentquo_hash': 11377, 'autcentquo_nilpotent': False, 'autcentquo_order': 960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_6\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [2, 6, 2], [2, 12, 2], [2, 20, 1], [3, 2, 1], [4, 2, 2], [4, 4, 1], [4, 6, 2], [4, 20, 1], [4, 60, 2], [5, 2, 2], [6, 2, 1], [6, 4, 1], [6, 8, 1], [6, 40, 1], [8, 10, 2], [8, 20, 1], [8, 60, 2], [10, 2, 2], [10, 4, 6], [10, 12, 12], [12, 2, 2], [12, 4, 1], [12, 8, 1], [12, 40, 1], [15, 4, 2], [20, 4, 8], [20, 12, 4], [24, 20, 4], [30, 4, 2], [30, 8, 6], [60, 4, 4], [60, 8, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '480.1123', 'commutator_count': 1, 'commutator_label': '60.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 9542, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 6, 1, 2], [2, 12, 1, 2], [2, 20, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 6, 1, 2], [4, 20, 1, 1], [4, 60, 1, 2], [5, 2, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 8, 1, 1], [6, 40, 1, 1], [8, 10, 2, 1], [8, 20, 1, 1], [8, 60, 1, 2], [10, 2, 2, 1], [10, 4, 2, 1], [10, 4, 4, 1], [10, 12, 2, 2], [10, 12, 4, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 8, 1, 1], [12, 40, 1, 1], [15, 4, 2, 1], [20, 4, 2, 2], [20, 4, 4, 1], [20, 12, 2, 2], [24, 20, 2, 2], [30, 4, 2, 1], [30, 8, 2, 1], [30, 8, 4, 1], [60, 4, 4, 1], [60, 8, 2, 1], [60, 8, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 16248960, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, -1, 4]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '240.202', 'hash': 9542, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 12, 10], 'inner_gens': [[1, 2, 76, 144], [1, 2, 44, 96], [25, 58, 4, 864], [49, 2, 196, 96]], 'inner_hash': 1123, 'inner_nilpotent': False, 'inner_order': 480, 'inner_split': True, 'inner_tex': 'D_{30}:C_2^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 32, 'irrQ_dim': 64, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 44], [4, 24], [8, 6]], 'label': '960.9542', 'linC_count': 128, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 64, 'linQ_dim': 14, 'linQ_dim_count': 64, 'linR_count': 32, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C60.C2^4', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 44, 'number_characteristic_subgroups': 76, 'number_conjugacy_classes': 90, 'number_divisions': 52, 'number_normal_subgroups': 136, 'number_subgroup_autclasses': 364, 'number_subgroup_classes': 516, 'number_subgroups': 2792, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 63], [3, 2], [4, 160], [5, 4], [6, 54], [8, 160], [10, 172], [12, 56], [15, 8], [20, 80], [24, 80], [30, 56], [60, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 480, 0], 'outer_gens': [[1, 2, 68, 96], [1, 2, 52, 96], [25, 482, 4, 96], [1, 2, 4, 288]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [5040, 362880, 120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 10], [8, 9], [16, 4], [32, 1]], 'representations': {'PC': {'code': 3979068386868250292594582861399756575104547551448989715, 'gens': [1, 2, 3, 7], 'pres': [8, 2, 2, 2, 2, 2, 3, 2, 5, 384, 1826, 538, 66, 1795, 1419, 91, 1612, 116, 1549, 8070, 12118, 166, 12311]}, 'Perm': {'d': 24, 'gens': [26120790521626347994344, 58725434311664498340481, 85897012702764369254400, 112422500799302284778880, 137588906839382955129600, 165442458588883993440000, 3, 6504]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{60}.C_2^4', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
