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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1200.923', 'ambient_counter': 923, 'ambient_order': 1200, 'ambient_tex': 'C_{10}^2.D_6', 'central': True, 'central_factor': False, 'centralizer_order': 1200, 'characteristic': True, 'core_order': 10, 'counter': 65, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1200.923.120.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '120.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '120.46', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 46, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 120, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times D_{30}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '10.2', 'subgroup_hash': 2, 'subgroup_order': 10, 'subgroup_tex': 'C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1200.923', 'aut_centralizer_order': 23040, 'aut_label': '120.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 23040, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['24.a1', '40.a1', '60.a1', '60.f1'], 'contains': ['240.a1', '600.a1'], 'core': '120.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2235, 7652, 6579, 1910], 'generators': [20, 8], 'label': '1200.923.120.a1', 'mobius_quo': 1, 'mobius_sub': -120, 'normal_closure': '120.a1', 'normal_contained_in': ['24.a1', '40.a1', '60.a1'], 'normal_contains': ['240.a1', '600.a1'], 'normalizer': '1.a1', 'old_label': '120.a1', 'projective_image': '120.46', 'quotient_action_image': '1.1', 'quotient_action_kernel': '120.46', 'quotient_action_kernel_order': 120, 'quotient_fusion': None, 'short_label': '120.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [7]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', '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, 1], [5, 1, 4], [10, 1, 4]], 'center_label': '10.2', 'center_order': 10, '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', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 10]], 'label': '10.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C10', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 10, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[7]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2]], 'representations': {'PC': {'code': 83, 'gens': [1], 'pres': [2, -2, -5, 4]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16717348]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504]}, 'Perm': {'d': 7, 'gens': [720, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}', 'transitive_degree': 10, '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': '80.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 60, 'aut_gen_orders': [4, 4, 12, 2, 60], 'aut_gens': [[1, 2, 40], [621, 518, 280], [21, 843, 301], [600, 723, 541], [620, 1039, 1061], [621, 646, 661]], 'aut_group': None, 'aut_hash': 7196069784989913445, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 92160, 'aut_permdeg': 124, 'aut_perms': [613087717892375850752684942710355862099338357228515441659327092725294904700456059530877763118468863369906039353991907673765971614757809033250984991704816578808894366539264049030022252479643153974989023000897, 772081329358186471083211497774826572543597597844954429285799219407805580415012049317776749287259308249163482906964859472531089839405259785012429638021618660419636343459474422297359901548029458540282145612043, 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1], [30, 2, 24, 2], [30, 2, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3.(C_{30}\\times A_4).C_2^4.C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '768.1090064', 'autcent_hash': 1090064, 'autcent_nilpotent': False, 'autcent_order': 768, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times C_2^3:S_4', '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, 7], [3, 2, 1], [4, 15, 8], [5, 1, 4], [5, 2, 10], [6, 2, 7], [10, 1, 28], [10, 2, 70], [15, 2, 24], [20, 15, 32], [30, 2, 168]], 'center_label': '40.14', 'center_order': 40, 'central_product': True, 'central_quotient': '30.3', 'commutator_count': 1, 'commutator_label': '15.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 923, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1], ['60.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 15, 2, 4], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [6, 2, 1, 7], [10, 1, 4, 7], [10, 2, 2, 7], [10, 2, 4, 14], [15, 2, 4, 2], [15, 2, 8, 2], [20, 15, 8, 4], [30, 2, 4, 14], [30, 2, 8, 14]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5208, 'exponent': 60, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '600.203', 'hash': 923, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [1, 2, 15], 'inner_gens': [[1, 2, 40], [1, 2, 1160], [1, 82, 40]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 30, 'inner_split': True, 'inner_tex': 'D_{15}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 280]], 'label': '1200.923', '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': 'C10^2.D6', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 360, 'number_divisions': 88, 'number_normal_subgroups': 150, 'number_subgroup_autclasses': 84, 'number_subgroup_classes': 280, 'number_subgroups': 784, 'old_label': None, 'order': 1200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 120], [5, 24], [6, 14], [10, 168], [15, 48], [20, 480], [30, 336]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [4, 4, 2, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[21, 14, 440], [620, 27, 1041], [1, 622, 440], [621, 7, 1141]], 'outer_group': None, 'outer_hash': 4658869075538623801, 'outer_nilpotent': False, 'outer_order': 3072, 'outer_permdeg': 16, 'outer_perms': [1321086780857, 8034294199697, 2877362616367, 1676629353720], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:A_4.C_2^2.C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 16], [8, 36], [16, 16]], 'representations': {'PC': {'code': 19748055474384388502267198537927, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -5, -2, -3, -5, 36, 58, 20311, 102, 23532, 166, 23533]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [7187425, 287521, 6612431, 289675, 288949, 12362371, 3294907]}, 'Perm': {'d': 21, 'gens': [2446777753349897670, 289, 5329, 5250034053358617600, 288, 3991680, 7665149074781721600]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2.D_6', 'transitive_degree': 240, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [3, 3, 6, 12, 2, 2, 2, 6, 60], 'aut_gens': [[1, 2, 4], [60, 3, 5], [1, 42, 4], [60, 83, 77], [61, 83, 52], [1, 62, 4], [60, 63, 65], [1, 3, 4], [61, 62, 105], [1, 78, 5]], 'aut_group': '2880.hl', 'aut_hash': 799193401477229059, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2880, 'aut_permdeg': 12, 'aut_perms': [11, 1440, 7984211, 10887257, 7, 5, 23, 124, 119792170], 'aut_phi_ratio': 90.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 15, 4, 1], [3, 2, 1, 1], [5, 2, 2, 1], [6, 2, 3, 1], [10, 2, 6, 1], [15, 2, 4, 1], [30, 2, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times F_5\\times S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', '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, 3], [2, 15, 4], [3, 2, 1], [5, 2, 2], [6, 2, 3], [10, 2, 6], [15, 2, 4], [30, 2, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '30.3', 'commutator_count': 1, 'commutator_label': '15.1', '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': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['30.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 15, 1, 4], [3, 2, 1, 1], [5, 2, 2, 1], [6, 2, 1, 3], [10, 2, 2, 3], [15, 2, 4, 1], [30, 2, 4, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 30, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '120.46', 'hash': 46, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [1, 2, 15], 'inner_gens': [[1, 2, 4], [1, 2, 116], [1, 10, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 30, 'inner_split': True, 'inner_tex': 'D_{15}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 28]], 'label': '120.46', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 6, 'linQ_dim': 6, 'linQ_dim_count': 6, 'linR_count': 48, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D30', '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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 36, 'number_divisions': 20, 'number_normal_subgroups': 31, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 64, 'number_subgroups': 284, 'old_label': None, 'order': 120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 63], [3, 2], [5, 4], [6, 6], [10, 12], [15, 8], [30, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 4, 4, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 44], [1, 63, 44], [1, 3, 28], [1, 2, 52], [1, 3, 45], [60, 3, 45]], 'outer_group': '96.186', 'outer_hash': 186, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [7, 11527, 16697, 22, 15967, 6607], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 4], [8, 4]], 'representations': {'PC': {'code': 10701074178256903, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -3, -5, 877, 42, 1128, 78, 1209]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [91793133226635161, 24992906222183252, 108378803088313658]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 32682189518, 106616128025, 132535578350, 87392415512]}, 'Perm': {'d': 12, 'gens': [3669126, 720, 721, 30, 47214720]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_2\\times D_{30}', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
