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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1458.122', 'ambient_counter': 122, 'ambient_order': 1458, 'ambient_tex': 'C_2\\times C_3^3:\\He_3', 'central': True, 'central_factor': False, 'centralizer_order': 1458, 'characteristic': True, 'core_order': 2, 'counter': 27, 'cyclic': True, 'direct': True, 'hall': 2, 'label': '1458.122.729.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '729.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '729.122', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 122, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 729, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^3:\\He_3', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '1458.122', 'aut_centralizer_order': 221079456, 'aut_label': '729.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 221079456, 'centralizer': '1.a1', 'complements': ['2.a1'], 'conjugacy_class_count': 1, 'contained_in': ['243.a1', '243.b1'], 'contains': ['1458.a1'], 'core': '729.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3898, 5441, 6674, 7476], 'generators': [729], 'label': '1458.122.729.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '729.a1', 'normal_contained_in': ['243.a1'], 'normal_contains': ['1458.a1'], 'normalizer': '1.a1', 'old_label': '729.a1', 'projective_image': '729.122', 'quotient_action_image': '1.1', 'quotient_action_kernel': '729.122', 'quotient_action_kernel_order': 729, 'quotient_fusion': None, 'short_label': '729.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': '54.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 936, 'aut_gen_orders': [24, 6], 'aut_gens': [[1, 3, 9, 27, 81, 243], [50, 142, 17, 675, 648, 837], [530, 118, 1179, 702, 189, 459]], 'aut_group': None, 'aut_hash': 1691416334942160361, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 221079456, 'aut_permdeg': 728, 'aut_perms': 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'aut_phi_ratio': 454896.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 26, 1], [3, 9, 78, 1], [6, 1, 26, 1], [6, 9, 78, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^9.C_2.\\SL(3,3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': None, 'autcent_hash': 5585183472077698458, 'autcent_nilpotent': True, 'autcent_order': 19683, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^9', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 312, 'autcentquo_group': '11232.a', 'autcentquo_hash': 778507202365856770, 'autcentquo_nilpotent': False, 'autcentquo_order': 11232, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(3,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 26], [3, 9, 78], [6, 1, 26], [6, 9, 78]], 'center_label': '54.15', 'center_order': 54, 'central_product': True, 'central_quotient': '27.5', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 122, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['729.122', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 13], [3, 9, 2, 39], [6, 1, 2, 13], [6, 9, 2, 39]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 7, 'exponent': 6, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '54.15', 'hash': 122, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 3, 1, 1, 1], 'inner_gens': [[1, 30, 576, 27, 81, 243], [55, 3, 225, 27, 81, 243], [1135, 111, 9, 27, 81, 243], [1, 3, 9, 27, 81, 243], [1, 3, 9, 27, 81, 243], [1, 3, 9, 27, 81, 243]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': True, 'inner_tex': 'C_3^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 54], [3, 156]], 'label': '1458.122', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, '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': 'C2*C3^3:He3', 'ngens': 7, 'nilpotency_class': 2, 'nilpotent': True, '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 210, 'number_divisions': 106, 'number_normal_subgroups': 188, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 2138, 'number_subgroups': 7910, 'old_label': None, 'order': 1458, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1], [3, 728], [6, 728]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 936, 'outer_gen_orders': [13, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[1132, 159, 559, 1188, 108, 918], [1104, 1070, 2, 1080, 81, 1404]], 'outer_group': '8188128.a', 'outer_hash': 3919716516049662436, 'outer_nilpotent': False, 'outer_order': 8188128, 'outer_permdeg': 39, 'outer_perms': [14990510986901463586458345836257753478290457941, 12839155308017292004532594876288516613659291146], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_3^6.\\GL(3,3)', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 29, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 26], [6, 78]], 'representations': {'PC': {'code': 2556096088282703097981, 'gens': [1, 2, 3, 4, 5, 6], 'pres': [7, -3, -3, -3, -3, -3, -2, -3, 421, 12098, 1584, 124]}, 'Perm': {'d': 29, 'gens': [418830873492106528486411988, 871215365496455591836964130, 1258997818235143032893629898, 1692715876881073278513494776, 195888, 1692715876881487643634013488, 304888344611713860501504000000]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_3^3:\\He_3', 'transitive_degree': 486, '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': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 936, 'aut_gen_orders': [26, 26], 'aut_gens': [[1, 3, 9, 27, 81, 243], [8, 226, 161, 189, 621, 270], [449, 628, 188, 594, 675, 540]], 'aut_group': None, 'aut_hash': 1691416334942160361, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 221079456, 'aut_permdeg': 702, 'aut_perms': 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'aut_phi_ratio': 454896.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 26, 1], [3, 9, 78, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^9.C_2.\\SL(3,3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': None, 'autcent_hash': 5585183472077698458, 'autcent_nilpotent': True, 'autcent_order': 19683, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^9', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 312, 'autcentquo_group': '11232.a', 'autcentquo_hash': 778507202365856770, 'autcentquo_nilpotent': False, 'autcentquo_order': 11232, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(3,3)', 'cc_stats': [[1, 1, 1], [3, 1, 26], [3, 9, 78]], 'center_label': '27.5', 'center_order': 27, 'central_product': False, 'central_quotient': '27.5', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 122, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 13], [3, 9, 2, 39]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '27.5', 'hash': 122, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 3, 1, 1, 1], 'inner_gens': [[1, 165, 495, 27, 81, 243], [82, 3, 63, 27, 81, 243], [244, 30, 9, 27, 81, 243], [1, 3, 9, 27, 81, 243], [1, 3, 9, 27, 81, 243], [1, 3, 9, 27, 81, 243]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': True, 'inner_tex': 'C_3^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 27], [3, 78]], 'label': '729.122', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, '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': 'C3^3:He3', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 105, 'number_divisions': 53, 'number_normal_subgroups': 94, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 1069, 'number_subgroups': 3955, 'old_label': None, 'order': 729, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 728]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 936, 'outer_gen_orders': [13, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[708, 504, 667, 243, 27, 108], [427, 97, 558, 513, 567, 540]], 'outer_group': '8188128.a', 'outer_hash': 3919716516049662436, 'outer_nilpotent': False, 'outer_order': 8188128, 'outer_permdeg': 39, 'outer_perms': [11308423069534071372522618867973236887686001593, 19323119229540747208699608790914495728947627942], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_3^6.\\GL(3,3)', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 27, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13], [6, 39]], 'representations': {'PC': {'code': 10032148120672, 'gens': [1, 2, 3, 4, 5, 6], 'pres': [6, -3, 3, 3, -3, 3, 3, 1981, 8912, 386]}, 'Perm': {'d': 27, 'gens': [418830873492106528486411988, 871215365496455591836964130, 1258997818235143032893629898, 1692715876881073278513494776, 195888, 1692715876881487643634013488]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:\\He_3', 'transitive_degree': 243, 'wreath_data': None, 'wreath_product': False}
