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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1836.48', 'ambient_counter': 48, 'ambient_order': 1836, 'ambient_tex': '\\He_3:D_{34}', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 9, 'counter': 57, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1836.48.51.a1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '51.a1.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': 51, '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': '36.12', 'subgroup_hash': 12, 'subgroup_order': 36, 'subgroup_tex': 'C_6\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1836.48', 'aut_centralizer_order': 48, 'aut_label': '51.a1', 'aut_quo_index': None, 'aut_stab_index': 204, 'aut_weyl_group': '12.4', 'aut_weyl_index': 9792, 'centralizer': '306.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1.b1', '17.a1.a1'], 'contains': ['102.a1.b1', '102.b1.b1', '102.c1.b1', '153.a1.a1', '153.b1.b1'], 'core': '204.a1.b1', 'coset_action_label': None, 'count': 51, 'diagramx': [7018, -1, 9339, -1, 6401, -1, 1571, -1], 'generators': [1, 6, 12, 4], 'label': '1836.48.51.a1.b1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '51.a1.b1', 'old_label': '51.a1.b1', 'projective_image': '612.26', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '51.a1.b1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 2, 6], 'aut_gens': [[1, 6], [5, 6], [1, 30], [31, 6]], 'aut_group': '24.14', 'aut_hash': 14, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [7, 127, 856], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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], [2, 3, 2], [3, 1, 2], [3, 2, 3], [6, 1, 2], [6, 2, 3], [6, 3, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, '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', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.12', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 30], [13, 6]], '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': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 12], [2, 6]], 'label': '36.12', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 7, 'linQ_dim': 4, 'linQ_dim_count': 7, 'linR_count': 7, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6*S3', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 18, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 36, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 8], [6, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[19, 6], [5, 6]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 2]], 'representations': {'PC': {'code': 403784533, 'gens': [1, 3], 'pres': [4, -2, -3, -2, -3, 8, 362, 34, 387]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858573, 35931481]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1718]}, 'Perm': {'d': 8, 'gens': [720, 1, 30, 5760]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times S_3', 'transitive_degree': 12, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 816, 'aut_gen_orders': [68, 16, 8], 'aut_gens': [[1, 2, 12, 36], [1245, 706, 12, 56], [645, 918, 24, 1664], [629, 1498, 24, 308]], 'aut_group': None, 'aut_hash': 6390200105259800102, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 117504, 'aut_permdeg': 26, 'aut_perms': [113110817189280637989457320, 125236288929195090822374959, 361649361151268990766837789], 'aut_phi_ratio': 204.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 17, 1, 1], [2, 153, 1, 1], [3, 1, 2, 1], [3, 6, 4, 1], [6, 9, 2, 1], [6, 17, 2, 1], [6, 102, 4, 1], [6, 153, 2, 1], [17, 2, 8, 1], [34, 18, 8, 1], [51, 2, 16, 1], [51, 12, 32, 1], [102, 18, 16, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSU(3,2).C_{51}.C_8.C_2^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 816, 'autcentquo_group': None, 'autcentquo_hash': 6390200105259800102, 'autcentquo_nilpotent': False, 'autcentquo_order': 117504, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PSU(3,2).C_{51}.C_8.C_2^2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 17, 1], [2, 153, 1], [3, 1, 2], [3, 6, 4], [6, 9, 2], [6, 17, 2], [6, 102, 4], [6, 153, 2], [17, 2, 8], [34, 18, 8], [51, 2, 16], [51, 12, 32], [102, 18, 16]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '612.26', 'commutator_count': 1, 'commutator_label': '459.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '17.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 48, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['34.1', 1], ['54.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 17, 1, 1], [2, 153, 1, 1], [3, 1, 2, 1], [3, 6, 1, 4], [6, 9, 2, 1], [6, 17, 2, 1], [6, 102, 1, 4], [6, 153, 2, 1], [17, 2, 8, 1], [34, 18, 8, 1], [51, 2, 16, 1], [51, 12, 8, 4], [102, 18, 16, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 20412, 'exponent': 102, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 17], 'factors_of_order': [2, 3, 17], 'faithful_reps': [[6, 0, 32]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '612.26', 'hash': 48, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 102, 'inner_gen_orders': [2, 6, 1, 51], 'inner_gens': [[1, 10, 12, 1260], [5, 2, 12, 600], [1, 2, 12, 36], [613, 1310, 12, 36]], 'inner_hash': 26, 'inner_nilpotent': False, 'inner_order': 612, 'inner_split': False, 'inner_tex': 'C_{51}:D_6', 'inner_used': [1, 2, 4], 'irrC_degree': 6, 'irrQ_degree': 96, 'irrQ_dim': 96, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 24], [3, 8], [4, 32], [6, 32]], 'label': '1836.48', 'linC_count': 128, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 8, 'linQ_dim': 22, 'linQ_dim_count': 8, 'linR_count': 64, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'He3:D34', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 15, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 100, 'number_divisions': 24, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 110, 'number_subgroups': 2014, 'old_label': None, 'order': 1836, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 179], [3, 26], [6, 766], [17, 16], [34, 144], [51, 416], [102, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 24, 2, 2], 'outer_gen_pows': [0, 6, 1, 1], 'outer_gens': [[1, 1230, 24, 656], [1, 1230, 12, 1492], [1, 1230, 12, 1192], [1, 1258, 12, 56]], 'outer_group': '192.958', 'outer_hash': 958, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 12, 'outer_perms': [362880, 7263513, 40279680, 127370880], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_8\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [6, 4], [16, 2], [32, 4], [96, 2]], 'representations': {'PC': {'code': 63105695830042829760471305814854865183, 'gens': [1, 2, 4, 5], 'pres': [6, -2, -2, -3, -3, 3, -17, 121, 31, 146, 37804, 9010, 376, 118, 31115]}, 'Perm': {'d': 26, 'gens': [1313941673647, 1266904769224087977984000, 1890778703387213832192000, 16803971573452874883072000, 23550673235293, 32966021836805616599040000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\He_3:D_{34}', 'transitive_degree': 153, 'wreath_data': None, 'wreath_product': False}
