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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1568.629', 'ambient_counter': 629, 'ambient_order': 1568, 'ambient_tex': 'C_{28}.D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 392, 'characteristic': False, 'core_order': 1, 'counter': 120, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1568.629.112.f1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '112.f1.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': 112, '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': '14.2', 'subgroup_hash': 2, 'subgroup_order': 14, 'subgroup_tex': 'C_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1568.629', 'aut_centralizer_order': 224, 'aut_label': '112.f1', 'aut_quo_index': None, 'aut_stab_index': 12, 'aut_weyl_group': '6.2', 'aut_weyl_index': 2688, 'centralizer': '4.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.b1.a1', '56.g1.b1'], 'contains': ['224.c1.b1', '784.b1.a1'], 'core': '1568.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [1456, -1, 9310, -1, 1794, -1, 1693, -1], 'generators': [1204, 1016], 'label': '1568.629.112.f1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.a1.a1', 'old_label': '112.f1.b1', 'projective_image': '1568.629', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '112.f1.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': '14.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [3]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', '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], [7, 1, 6], [14, 1, 6]], 'center_label': '14.2', 'center_order': 14, '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', '7.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 14, 'eulerian_function': 1, 'exponent': 14, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '14.2', 'hash': 2, 'hyperelementary': 14, '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': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 14]], 'label': '14.2', 'linC_count': 6, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C14', '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': 14, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 14, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [7, 6], [14, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [6, 2]], 'representations': {'PC': {'code': 185, 'gens': [1], 'pres': [2, -2, -7, 4]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [75621406240170985]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [351, 2064]}, 'Perm': {'d': 9, 'gens': [40320, 4320]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [14], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{14}', 'transitive_degree': 14, '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': '56.13', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [4, 6, 2, 6, 6, 6, 28], 'aut_gens': [[1, 2, 56], [805, 1230, 56], [785, 426, 616], [785, 1230, 1512], [793, 10, 840], [817, 838, 616], [29, 1214, 616], [833, 422, 56]], 'aut_group': None, 'aut_hash': 1967663635867252092, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 42, 'aut_perms': [670706633111263220808599153959242645917753761761845, 1061993977554914046748080290020564761729185854996956, 1250699976652069024487594308417267411783407475429805, 886677271424931344666738722001631331395885091923999, 617697458494758646254918516265601438337308264072527, 368246550577422001795206237740265173386024095829019, 1232750973253777482577001922311526781915699685623259], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 28, 1, 1], [4, 2, 1, 2], [4, 28, 1, 1], [4, 28, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 18, 1], [8, 4, 2, 1], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 2, 18, 1], [14, 4, 3, 1], [14, 4, 18, 1], [14, 28, 6, 1], [28, 2, 6, 3], [28, 2, 36, 1], [28, 4, 3, 1], [28, 4, 18, 1], [28, 28, 6, 1], [28, 28, 12, 1], [56, 4, 12, 2], [56, 4, 72, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_7.(C_6^2\\times D_4).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '48.52', 'autcent_hash': 52, 'autcent_nilpotent': True, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '336.216', 'autcentquo_hash': 216, 'autcentquo_nilpotent': False, 'autcentquo_order': 336, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 28, 1], [4, 2, 2], [4, 28, 3], [7, 1, 6], [7, 2, 21], [8, 4, 2], [14, 1, 6], [14, 2, 27], [14, 4, 21], [14, 28, 6], [28, 2, 54], [28, 4, 21], [28, 28, 18], [56, 4, 96]], 'center_label': '14.2', 'center_order': 14, 'central_product': True, 'central_quotient': '112.29', 'commutator_count': 1, 'commutator_label': '28.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 629, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['224.104', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 28, 1, 1], [4, 2, 1, 2], [4, 28, 1, 3], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 6, 3], [8, 4, 1, 2], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 6, 4], [14, 4, 3, 1], [14, 4, 6, 3], [14, 28, 6, 1], [28, 2, 6, 3], [28, 2, 12, 3], [28, 4, 3, 1], [28, 4, 6, 3], [28, 28, 6, 3], [56, 4, 6, 4], [56, 4, 12, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 76608, 'exponent': 56, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 0, 36]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '392.42', 'hash': 629, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 28, 2], 'inner_gens': [[1, 54, 56], [789, 2, 840], [1, 786, 56]], 'inner_hash': 29, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': True, 'inner_tex': 'C_2\\times D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 56], [2, 182], [4, 49]], 'label': '1568.629', 'linC_count': 36, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 44, 'linQ_dim': 20, 'linQ_dim_count': 44, 'linR_count': 18, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C28.D28', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 287, 'number_divisions': 50, 'number_normal_subgroups': 58, 'number_subgroup_autclasses': 109, 'number_subgroup_classes': 150, 'number_subgroups': 606, 'old_label': None, 'order': 1568, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 31], [4, 88], [7, 48], [8, 8], [14, 312], [28, 696], [56, 384]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6, 6], 'outer_gen_pows': [0, 0, 0, 56], 'outer_gens': [[1, 2, 840], [1, 814, 56], [1, 2, 952], [1, 430, 56]], 'outer_group': '144.197', 'outer_hash': 197, 'outer_nilpotent': True, 'outer_order': 144, 'outer_permdeg': 14, 'outer_perms': [6227020800, 5040, 363024, 39916803], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 7], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 1], [6, 12], [12, 16], [24, 8], [48, 3]], 'representations': {'PC': {'code': 13860378091811011953205171000208735238461, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -7, -2, -2, -7, 757, 36, 17558, 58, 23299, 2768, 14711, 102, 124]}, 'Perm': {'d': 30, 'gens': [8865156130201232820085425393643, 3079988947639, 4491426195058, 18621225732230635889760473088000, 5670545596524, 7177921676593, 27790492704163172295979290624000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 14], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}.D_{28}', 'transitive_degree': 112, 'wreath_data': None, 'wreath_product': False}
