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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '288.960', 'ambient_counter': 960, 'ambient_order': 288, 'ambient_tex': 'D_4:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 144, 'characteristic': False, 'core_order': 4, 'counter': 137, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '288.960.72.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '72.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '72.46', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 46, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 72, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3\\times D_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '288.960', 'aut_centralizer_order': 576, 'aut_label': '72.a1', 'aut_quo_index': 4, 'aut_stab_index': 2, 'aut_weyl_group': '2.1', 'aut_weyl_index': 1152, 'centralizer': '2.c1', 'complements': [], 'conjugacy_class_count': 2, 'contained_in': ['24.a1', '24.f1', '36.a1', '36.b1', '36.c1', '36.d1', '36.e1'], 'contains': ['144.a1', '144.b1'], 'core': '72.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [4639, 2349, 4235, 8728], 'generators': [12, 144], 'label': '288.960.72.a1', 'mobius_quo': 0, 'mobius_sub': -72, 'normal_closure': '72.a1', 'normal_contained_in': ['24.a1', '36.a1'], 'normal_contains': ['144.a1'], 'normalizer': '1.a1', 'old_label': '72.a1', 'projective_image': '144.192', 'quotient_action_image': '2.1', 'quotient_action_kernel': '36.10', 'quotient_action_kernel_order': 36, 'quotient_fusion': None, 'short_label': '72.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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, 3]], 'center_label': '4.2', 'center_order': 4, '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', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 4, 2, 6, 6, 4, 2], 'aut_gens': [[1, 2, 4, 24], [1, 2, 148, 24], [1, 2, 4, 168], [1, 146, 4, 24], [1, 2, 76, 168], [154, 193, 204, 80], [145, 146, 4, 24], [49, 2, 4, 24], [145, 10, 148, 168], [1, 146, 92, 24], [145, 146, 164, 120]], 'aut_group': '2304.eq', 'aut_hash': 8462164704074420658, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2304, 'aut_permdeg': 16, 'aut_perms': [19709367160617, 20433899184117, 1959925760, 5955433117410, 20269575624449, 724855395804, 436425300870, 3680482736, 15787900751471, 20172082331507], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [2, 6, 2, 1], [2, 9, 2, 1], [2, 18, 2, 1], [3, 2, 2, 1], [3, 4, 1, 1], [4, 2, 1, 1], [4, 3, 4, 1], [4, 6, 4, 1], [4, 18, 1, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 4, 4, 1], [6, 8, 2, 1], [6, 12, 2, 1], [12, 4, 2, 1], [12, 6, 4, 1], [12, 8, 1, 1], [12, 12, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2:D_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '144.186', 'autcentquo_hash': 186, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [2, 6, 2], [2, 9, 2], [2, 18, 2], [3, 2, 2], [3, 4, 1], [4, 2, 1], [4, 3, 4], [4, 6, 4], [4, 18, 1], [6, 2, 2], [6, 4, 5], [6, 8, 2], [6, 12, 2], [12, 4, 2], [12, 6, 4], [12, 8, 1], [12, 12, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '144.192', 'commutator_count': 1, 'commutator_label': '18.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 960, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [2, 6, 1, 2], [2, 9, 1, 2], [2, 18, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 1, 1], [4, 3, 2, 2], [4, 6, 1, 4], [4, 18, 1, 1], [6, 2, 1, 2], [6, 4, 1, 5], [6, 8, 1, 2], [6, 12, 1, 2], [12, 4, 1, 2], [12, 6, 2, 2], [12, 8, 1, 1], [12, 12, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 851760, 'exponent': 12, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '144.192', 'hash': 960, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6, 6], 'inner_gens': [[1, 2, 148, 120], [1, 2, 164, 24], [145, 154, 4, 168], [193, 2, 148, 24]], 'inner_hash': 192, 'inner_nilpotent': False, 'inner_order': 144, 'inner_split': True, 'inner_tex': 'D_6^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 20], [4, 8], [8, 1]], 'label': '288.960', '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': 'D4:S3^2', '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': 21, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 45, 'number_divisions': 41, 'number_normal_subgroups': 110, 'number_subgroup_autclasses': 154, 'number_subgroup_classes': 355, 'number_subgroups': 1346, 'old_label': None, 'order': 288, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 71], [3, 8], [4, 56], [6, 64], [12, 88]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 72, 0], 'outer_gens': [[2, 1, 204, 224], [145, 2, 20, 120], [1, 2, 92, 24], [1, 2, 20, 168]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 7, 127, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 6], [8, 3]], 'representations': {'PC': {'code': 946561883372431746463900672940553, 'gens': [1, 2, 3, 5], 'pres': [7, -2, -2, -2, -3, -2, -2, -3, 3110, 1731, 58, 234, 4204, 1488, 102, 10085, 124, 9414]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27301, 27019, 40966, 472517, 715966, 513019, 157325]}, 'Perm': {'d': 14, 'gens': [6314112526, 6354392206, 7158, 12316, 18498, 13971242880, 20122058880]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4:S_3^2', 'transitive_degree': 24, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 3, 3], 'aut_gens': [[1, 2, 12], [37, 46, 12], [6, 25, 40], [1, 38, 12], [1, 10, 60], [37, 38, 12], [9, 2, 12], [1, 26, 12]], 'aut_group': '288.889', 'aut_hash': 889, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 10, 'aut_perms': [726, 374423, 1, 16687, 7, 126000, 782040], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 4, 1], [2, 9, 2, 1], [3, 2, 2, 1], [3, 4, 1, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 6, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\wr C_2', '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': 12, 'autcentquo_group': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [3, 2, 2], [3, 4, 1], [6, 2, 2], [6, 4, 1], [6, 6, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '36.10', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 336, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '72.46', 'hash': 46, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 3], 'inner_gens': [[1, 10, 12], [5, 2, 60], [1, 26, 12]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 8], [4, 2]], 'label': '72.46', 'linC_count': 13, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 13, 'linQ_dim': 4, 'linQ_dim_count': 13, 'linR_count': 13, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*D6', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 18, 'number_divisions': 18, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 69, 'number_subgroups': 206, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 31], [3, 8], [6, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 38, 12], [42, 49, 40]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [6, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 2]], 'representations': {'PC': {'code': 1043840168412997, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -3, -2, -3, 101, 26, 122, 608, 58, 609]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780359, 26483311, 7115160]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 8101, 13286, 13933]}, 'Perm': {'d': 8, 'gens': [25, 720, 24, 5760, 3]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times D_6', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
