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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '264.26', 'ambient_counter': 26, 'ambient_order': 264, 'ambient_tex': 'C_{66}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 6, 'counter': 19, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '264.26.22.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '22.b1.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': 22, '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': '12.1', 'subgroup_hash': 1, 'subgroup_order': 12, 'subgroup_tex': 'C_3:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '264.26', 'aut_centralizer_order': 20, 'aut_label': '22.b1', 'aut_quo_index': None, 'aut_stab_index': 22, 'aut_weyl_group': '12.4', 'aut_weyl_index': 440, 'centralizer': '66.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.b1', '11.a1.a1'], 'contains': ['44.b1.a1', '66.b1.b1'], 'core': '44.b1.a1', 'coset_action_label': None, 'count': 11, 'diagramx': [8577, -1, 6218, -1, 7854, -1, 9665, -1], 'generators': [245, 2, 176], 'label': '264.26.22.b1.b1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '11.a1.a1', 'old_label': '22.b1.b1', 'projective_image': '132.9', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '22.b1.b1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 4], [1, 8], [11, 4]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 3, 2, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 1], [4, 3, 2], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, '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, 3, 2, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 12, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, -1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 8], [9, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.1', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:C4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 8, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 6], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3, 4]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3]], 'representations': {'PC': {'code': 3913, 'gens': [1, 3], 'pres': [3, -2, -2, -3, 6, 74]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20468488, 16858410]}, 'Lie': [{'d': 2, 'q': 7, 'gens': [2064, 1718, 690, 273], 'family': 'CSOPlus'}, {'d': 2, 'q': 5, 'gens': [537, 504, 427], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [169, 458]}, 'Perm': {'d': 7, 'gens': [129, 16, 840]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_4', 'transitive_degree': 12, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [20, 30, 10, 60, 30], 'aut_gens': [[1, 4], [135, 214], [249, 54], [137, 188], [117, 102], [65, 76]], 'aut_group': None, 'aut_hash': 3017774860523347021, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5280, 'aut_permdeg': 132, 'aut_perms': [79117083684770278281974879404225953382330106893062154257908055739483843802891833833843304972563075539655906380186088695982859858580225212702293846510389087746366153008677722821968835631337390581086858022261929659645827439081, 75411084350692281250927972817761066250787393778476748534557722605365238211092398299164324954585139598215011212521776493898604619584938711216942158173625673071597048321013501692165308768077896735566404123436037299898759260463, 55931574563573035198748287469623735764971573155741108596708456255999626283106132655258929871714880039068261040960367041828615789033316058076764121651820013865253816232964091929212299744953801671107316542076590867456342042521, 95530065899954599886949819082396320208820403147759780309945283469819652072422493557512750179125586081549990046716225027569753094070890068526427235946234900359396612999686265577205681638438454839823845371013241737478716107322, 83532114782973985030186893104702651742935078185381517948367821536659545740389568248662279501434913443759930082309440448141308054316011787948031403035515304612768338109037058571248382839447166031805453053925548562859090197366], 'aut_phi_ratio': 66.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 1, 1], [4, 33, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [11, 2, 5, 1], [22, 2, 5, 1], [22, 2, 10, 1], [33, 2, 10, 1], [66, 2, 10, 1], [66, 2, 20, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{66}.C_{10}.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '660.15', 'autcentquo_hash': 15, 'autcentquo_nilpotent': False, 'autcentquo_order': 660, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 33, 4], [6, 2, 3], [11, 2, 5], [22, 2, 15], [33, 2, 10], [66, 2, 30]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '66.3', 'commutator_count': 1, 'commutator_label': '33.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 26, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['132.3', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 33, 2, 2], [6, 2, 1, 3], [11, 2, 5, 1], [22, 2, 5, 3], [33, 2, 10, 1], [66, 2, 10, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 132, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '132.9', 'hash': 26, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [2, 33], 'inner_gens': [[1, 260], [9, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 66, 'inner_split': True, 'inner_tex': 'D_{33}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 64]], 'label': '264.26', 'linC_count': 120, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 6, 'linQ_dim': 14, 'linQ_dim_count': 8, 'linR_count': 20, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C66:C4', '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': 13, 'number_conjugacy_classes': 72, 'number_divisions': 18, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 32, 'number_subgroups': 164, 'old_label': None, 'order': 264, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 132], [6, 6], [11, 10], [22, 30], [33, 20], [66, 60]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [2, 2, 20], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[3, 172], [133, 172], [133, 102]], 'outer_group': '80.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 80, 'outer_permdeg': 11, 'outer_perms': [720, 3669840, 4354650], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_{10}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [10, 4], [20, 4]], 'representations': {'PC': {'code': 19077683717121662100019, 'gens': [1, 3], 'pres': [5, -2, -2, -2, -3, -11, 10, 3902, 42, 5123, 78, 6004]}, 'GLZN': {'d': 2, 'p': 44, 'gens': [1788885, 1959255, 1044979, 85361, 874974]}, 'Perm': {'d': 20, 'gens': [128403162022025647, 1307674368000, 256094948229120000, 6706022400, 4364893]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{66}:C_4', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
