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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '192.605', 'ambient_counter': 605, 'ambient_order': 192, 'ambient_tex': 'D_{12}.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 1, 'counter': 183, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '192.605.96.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '96.d1.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': 96, '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': '192.605', 'aut_centralizer_order': 768, 'aut_label': '96.d1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '1.1', 'aut_weyl_index': 3072, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.d1.b1', '48.b1.b1', '48.e1.a1', '48.i1.a1', '48.l1.e1', '48.l1.f1', '48.l1.g1', '48.l1.h1'], 'contains': ['192.a1.a1'], 'core': '192.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': None, 'generators': [104], 'label': '192.605.96.d1.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '48.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '96.d1.b1', 'projective_image': '192.605', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '96.d1.b1', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [12, 2, 4, 4, 2, 2], 'aut_gens': [[1, 2, 16, 32], [29, 62, 16, 40], [1, 106, 16, 32], [5, 102, 16, 56], [13, 182, 16, 184], [5, 86, 16, 160], [17, 158, 16, 184]], 'aut_group': '1024.dia', 'aut_hash': 5984332524656990092, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072, 'aut_permdeg': 56, 'aut_perms': [542652287606634630763197664997661472191358145620375444789444030770613022311, 165292242959323657230037266862127069745683115888006868958194293582843985325, 193036238576657690726083931863494149213747649272313923269688075794679768879, 682082125906495420887076188856433442117957748205302887003465209261679880560, 231125403607178121609894897764755329505391792014434848068427850410713636379, 444841064988347595043083579990003497473837772090049281603646543571636710594], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 12, 4, 1], [3, 2, 1, 1], [4, 2, 2, 1], [4, 4, 1, 1], [4, 8, 1, 2], [6, 2, 1, 3], [6, 4, 2, 1], [8, 12, 4, 1], [12, 4, 2, 2], [12, 8, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^7:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': None, 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': 48, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 12, 4], [3, 2, 1], [4, 2, 2], [4, 4, 1], [4, 8, 2], [6, 2, 3], [6, 4, 2], [8, 12, 4], [12, 4, 4], [12, 8, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '48.43', 'commutator_count': 2, 'commutator_label': '24.9', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 605, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 12, 1, 4], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 8, 1, 2], [6, 2, 1, 3], [6, 4, 1, 2], [8, 12, 2, 2], [12, 4, 1, 2], [12, 4, 2, 1], [12, 8, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 672, 'exponent': 24, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '24.14', 'hash': 605, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 1, 6], 'inner_gens': [[1, 6, 16, 48], [13, 2, 16, 176], [1, 2, 16, 32], [17, 82, 16, 32]], 'inner_hash': 43, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': None, 'inner_tex': 'C_6:D_4', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 18], [4, 7]], 'label': '192.605', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'D12.D4', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 20, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 33, 'number_divisions': 28, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': 100, 'number_subgroup_classes': 188, 'number_subgroups': 704, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 55], [3, 2], [4, 24], [6, 14], [8, 48], [12, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 4, 4, 2], 'outer_gen_pows': [0, 96, 0, 0, 0], 'outer_gens': [[1, 126, 16, 40], [29, 110, 16, 32], [5, 98, 16, 40], [1, 118, 16, 32], [17, 10, 16, 32]], 'outer_group': '64.202', 'outer_hash': 202, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 16, 'outer_perms': [19263838565063, 20661255014327, 12844489257690, 14274565877783, 6723627333817], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3:D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 9], [8, 1]], 'representations': {'PC': {'code': 33054227852618789581869970305, 'gens': [1, 2, 5, 6], 'pres': [7, -2, -2, -2, -2, -2, 2, -3, 56, 85, 36, 254, 58, 2021, 3708, 124, 3149]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038634252770949, 125101743816457766, 91793115019494933]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [14017, 13969, 158987, 318173, 14113, 13853, 179725]}, 'Perm': {'d': 15, 'gens': [87657658372, 186810624000, 479008758, 12568, 87657292800, 18498, 3991680]}}, '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': 'D_{12}.D_4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
