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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1944.3473', 'ambient_counter': 3473, 'ambient_order': 1944, 'ambient_tex': 'C_3^3.F_9', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 9, 'counter': 22, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1944.3473.54.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '54.b1', 'outer_equivalence': True, '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': 54, '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': '36.7', 'subgroup_hash': 7, 'subgroup_order': 36, 'subgroup_tex': 'C_3^2:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1944.3473', 'aut_centralizer_order': 24, 'aut_label': '54.b1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '144.182', 'aut_weyl_index': 216, 'centralizer': '324.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['18.a1', '27.a1'], 'contains': ['108.c1', '162.b1'], 'core': '216.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [1769, -1, 9039, -1], 'generators': [2, 4, 152, 72], 'label': '1944.3473.54.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1', 'old_label': '54.b1', 'projective_image': '1944.3473', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.b1', 'subgroup_fusion': None, 'weyl_group': '36.9'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 4, 3, 2, 6, 6], 'aut_gens': [[1, 4, 12], [3, 4, 12], [23, 28, 32], [25, 8, 24], [19, 16, 28], [9, 4, 12], [1, 32, 12], [19, 4, 12], [11, 20, 24]], 'aut_group': '864.4661', 'aut_hash': 4661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 11, 'aut_perms': [1, 10889689, 24054000, 20507935, 14605950, 2245249, 19496545, 16180705], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 4, 1], [4, 9, 2, 1], [6, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_3^2:\\GL(2,3)', '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': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 4], [4, 9, 2], [6, 2, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', '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': 7, '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, 4], [4, 9, 2, 1], [6, 2, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 28, 'exponent': 12, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '18.4', 'hash': 7, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 8, 24], [9, 4, 12], [25, 4, 12]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 8]], 'label': '36.7', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 6, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:C4', '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 12, 'number_divisions': 11, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 18, 'number_subgroups': 34, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 18], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 1, 1], 'outer_gens': [[3, 24, 8], [3, 4, 12], [1, 24, 28], [1, 32, 20], [1, 20, 16]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9]], 'representations': {'PC': {'code': 11576083, 'gens': [1, 3, 4], 'pres': [4, -2, -2, -3, -3, 8, 98, 387]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125124616718753647, 58415899998881273, 125101736062734304]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [11689448, 17625105, 28816400, 37924170]}, 'Perm': {'d': 10, 'gens': [41185, 1680, 403203, 3]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:C_4', 'transitive_degree': 36, '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': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [12, 8, 24, 4], 'aut_gens': [[1, 8, 24, 72, 216, 648], [1139, 88, 696, 80, 888, 1296], [713, 88, 480, 8, 48, 1296], [1265, 72, 24, 80, 1512, 648], [403, 144, 1752, 8, 432, 1296]], 'aut_group': None, 'aut_hash': 8814442550351613409, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 31104, 'aut_permdeg': 486, 'aut_perms': 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'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 4, 2, 1], [3, 4, 4, 1], [3, 24, 1, 1], [3, 24, 8, 1], [4, 81, 2, 1], [6, 18, 1, 1], [6, 36, 2, 1], [6, 36, 4, 1], [8, 243, 2, 2], [12, 162, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_3^2\\times \\He_3).C_8^2.C_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': 24, 'autcentquo_group': None, 'autcentquo_hash': 8814442550351613409, 'autcentquo_nilpotent': False, 'autcentquo_order': 31104, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3.C_3^4:C_2^2.C_4:D_4', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 1], [3, 4, 6], [3, 24, 9], [4, 81, 2], [6, 18, 1], [6, 36, 6], [8, 243, 4], [12, 162, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.3473', 'commutator_count': 1, 'commutator_label': '243.62', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3473, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 4, 1, 2], [3, 4, 2, 2], [3, 24, 1, 1], [3, 24, 2, 4], [4, 81, 2, 1], [6, 18, 1, 1], [6, 36, 1, 2], [6, 36, 2, 2], [8, 243, 4, 1], [12, 162, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 24, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 8]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '648.710', 'hash': 3473, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [8, 3, 3, 3, 3, 3], 'inner_gens': [[1, 160, 864, 88, 456, 1296], [89, 8, 24, 72, 216, 648], [457, 8, 24, 72, 1512, 648], [9, 8, 24, 72, 216, 648], [1129, 8, 672, 72, 216, 648], [1297, 8, 24, 72, 216, 648]], 'inner_hash': 3473, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': True, 'inner_tex': 'C_3^3.F_9', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [4, 4], [6, 4], [8, 9], [12, 8]], 'label': '1944.3473', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C3^3.F9', 'ngens': 8, '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': 14, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 33, 'number_divisions': 20, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 63, 'number_subgroup_classes': 144, 'number_subgroups': 2264, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 9], [3, 242], [4, 162], [6, 234], [8, 972], [12, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 8, 'outer_gen_orders': [2, 8], 'outer_gen_pows': [4, 0], 'outer_gens': [[3, 8, 888, 152, 456, 648], [1, 88, 24, 8, 216, 648]], 'outer_group': '16.8', 'outer_hash': 8, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [4761, 7602], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': '\\SD_{16}', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 33, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 5], [6, 2], [8, 1], [12, 1], [16, 4], [24, 4]], 'representations': {'PC': {'code': 99702897758858508457332253025251313008553975453765731103, 'gens': [1, 4, 5, 6, 7, 8], 'pres': [8, -2, -2, -2, -3, 3, 3, 3, -3, 16, 41, 5123, 267, 34564, 35052, 6980, 4229, 3469, 25542, 13454, 6070, 3566, 82951]}, 'Perm': {'d': 33, 'gens': [26623572949910681785411638872819889, 35034452734229988522576174545995456, 43522963801640179201739062199059200, 306954736513841819570653009619429040, 578545468616193147127084197783230640, 59, 304, 847849739698839490270833622484903040]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3.F_9', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
