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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '7200.dx', 'ambient_counter': 102, 'ambient_order': 7200, 'ambient_tex': 'S_6:C_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 240, 'characteristic': False, 'core_order': 1, 'counter': 129, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '7200.dx.3600.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '3600.a1.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': 3600, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7200.dx', 'aut_centralizer_order': None, 'aut_label': '3600.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '30.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['720.a1.a1', '1200.a1.a1', '1200.b1.a1', '1800.a1.a1', '1800.a2.a1'], 'contains': ['7200.a1.a1'], 'core': '7200.a1.a1', 'coset_action_label': None, 'count': 30, 'diagramx': [9030, -1, 2846, -1, 3656, -1, 3583, -1], 'generators': [2970682], 'label': '7200.dx.3600.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '10.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '30.a1.a1', 'old_label': '3600.a1.a1', 'projective_image': '7200.dx', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3600.a1.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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [20, 4, 8], 'aut_gens': [[194037282888, 1134023, 94363315200], [261578342810, 126736, 194035564800], [94363795094, 3118718, 367434144000], [261575893247, 1583281, 194035564800]], 'aut_group': '5760.db', 'aut_hash': 4554243923608446656, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5760, 'aut_permdeg': 70, 'aut_perms': [173639102378991849807104971111354233789157033965380637628499107782415269030327712960485229637853649, 513367532906672519583143717846731406831883053086185335890140513817795620644484523436529603184095498, 173639167458257439279959607258711339767062328741384046000894660034806888218400016113526174888735978], 'aut_phi_ratio': 3.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 30, 1, 1], [2, 36, 1, 1], [2, 45, 1, 1], [3, 80, 1, 1], [4, 90, 1, 2], [4, 180, 1, 1], [5, 1, 4, 1], [5, 144, 1, 1], [5, 144, 4, 1], [6, 240, 1, 1], [8, 180, 1, 2], [10, 30, 4, 1], [10, 36, 4, 1], [10, 45, 4, 1], [10, 144, 1, 1], [10, 144, 4, 1], [15, 80, 4, 1], [20, 90, 4, 2], [20, 180, 4, 1], [30, 240, 4, 1], [40, 180, 4, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_6:C_2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '1440.5841', 'autcentquo_hash': 5841, 'autcentquo_nilpotent': False, 'autcentquo_order': 1440, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_6:C_2', 'cc_stats': [[1, 1, 1], [2, 30, 1], [2, 36, 1], [2, 45, 1], [3, 80, 1], [4, 90, 2], [4, 180, 1], [5, 1, 4], [5, 144, 5], [6, 240, 1], [8, 180, 2], [10, 30, 4], [10, 36, 4], [10, 45, 4], [10, 144, 5], [15, 80, 4], [20, 90, 8], [20, 180, 4], [30, 240, 4], [40, 180, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '1440.5841', 'commutator_count': 1, 'commutator_label': '360.118', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '360.118'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 102, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['1440.5841', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 30, 1, 1], [2, 36, 1, 1], [2, 45, 1, 1], [3, 80, 1, 1], [4, 90, 1, 2], [4, 180, 1, 1], [5, 1, 4, 1], [5, 144, 1, 1], [5, 144, 4, 1], [6, 240, 1, 1], [8, 180, 1, 2], [10, 30, 4, 1], [10, 36, 4, 1], [10, 45, 4, 1], [10, 144, 1, 1], [10, 144, 4, 1], [15, 80, 4, 1], [20, 90, 4, 2], [20, 180, 4, 1], [30, 240, 4, 1], [40, 180, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2808, 'exponent': 120, 'exponents_of_order': [5, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[9, 0, 16], [10, 0, 8], [16, 0, 8], [20, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '7200.dx', 'hash': 6884383675079468252, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [10, 2, 1], 'inner_gens': [[194037282888, 2970682, 94363315200], [194038052536, 1134023, 94363315200], [194037282888, 1134023, 94363315200]], 'inner_hash': 5841, 'inner_nilpotent': False, 'inner_order': 1440, 'inner_split': True, 'inner_tex': 'S_6:C_2', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 18, 'irrep_stats': [[1, 20], [9, 20], [10, 10], [16, 10], [20, 5]], 'label': '7200.dx', 'linC_count': 16, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 13, 'linQ_degree_count': 16, 'linQ_dim': 13, 'linQ_dim_count': 16, 'linR_count': 32, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'S6:C10', 'ngens': 3, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 65, 'number_divisions': 26, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 132, 'number_subgroup_classes': 132, 'number_subgroups': 5446, 'old_label': None, 'order': 7200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 111], [3, 80], [4, 360], [5, 724], [6, 240], [8, 360], [10, 1164], [15, 320], [20, 1440], [30, 960], [40, 1440]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[261576508488, 1134023, 194035564800]], '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': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4], [9, 4], [10, 2], [16, 2], [20, 1], [36, 4], [40, 2], [64, 2], [80, 1]], 'representations': {'Perm': {'d': 15, 'gens': [194037282888, 1134023, 94363315200]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_6:C_{10}', 'transitive_degree': 50, 'wreath_data': None, 'wreath_product': False}
