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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '672.1172', 'ambient_counter': 1172, 'ambient_order': 672, 'ambient_tex': 'D_{42}:C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 84, 'counter': 12, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '672.1172.4.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.f1', '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': 4, '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': '168.54', 'subgroup_hash': 54, 'subgroup_order': 168, 'subgroup_tex': 'C_6\\times D_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '672.1172', 'aut_centralizer_order': 24, 'aut_label': '4.f1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '2016.dc', 'aut_weyl_index': 192, 'centralizer': '28.b1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['2.b1'], 'contains': ['8.e1', '8.f1', '12.f1', '28.f1'], 'core': '8.e1', 'coset_action_label': None, 'count': 8, 'diagramx': [2221, -1, 1665, -1], 'generators': [1, 336, 224, 2, 16], 'label': '672.1172.4.f1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1', 'old_label': '4.f1', 'projective_image': '168.16', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.f1', 'subgroup_fusion': None, 'weyl_group': '14.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 6], 'aut_gens': [[1, 2, 4], [1, 86, 20], [1, 74, 101], [84, 134, 45]], 'aut_group': '2016.dc', 'aut_hash': 2868159204135431305, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2016, 'aut_permdeg': 13, 'aut_perms': [3669724, 1037838251, 1049451609], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 7, 4, 1], [3, 1, 2, 1], [6, 1, 6, 1], [6, 7, 8, 1], [7, 2, 3, 1], [14, 2, 9, 1], [21, 2, 6, 1], [42, 2, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4\\times F_7', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '48.48', 'autcent_hash': 48, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 7, 4], [3, 1, 2], [6, 1, 6], [6, 7, 8], [7, 2, 3], [14, 2, 9], [21, 2, 6], [42, 2, 18]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 54, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 7, 1, 4], [3, 1, 2, 1], [6, 1, 2, 3], [6, 7, 2, 4], [7, 2, 3, 1], [14, 2, 3, 3], [21, 2, 6, 1], [42, 2, 6, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 728, 'exponent': 42, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '168.54', 'hash': 54, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [1, 2, 7], 'inner_gens': [[1, 2, 4], [1, 2, 52], [1, 122, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 36]], 'label': '168.54', 'linC_count': 288, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 12, 'linQ_dim': 8, 'linQ_dim_count': 12, 'linR_count': 36, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6*D14', '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 60, 'number_divisions': 24, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 64, 'number_subgroups': 196, 'old_label': None, 'order': 168, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 31], [3, 2], [6, 62], [7, 6], [14, 18], [21, 12], [42, 36]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [6, 6, 6, 6, 3, 3], 'outer_gen_pows': [0, 0, 0, 0, 2, 2], 'outer_gens': [[1, 86, 124], [85, 86, 76], [1, 2, 68], [1, 87, 20], [84, 2, 77], [1, 2, 76]], 'outer_group': '144.188', 'outer_hash': 188, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 9, 'outer_perms': [41070, 769, 31, 90769, 80688, 48], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [6, 4], [12, 4]], 'representations': {'PC': {'code': 509861979519585419, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -3, -7, 397, 42, 1048, 78, 1809]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [29071310, 7536919, 28836015, 33731060, 23068812]}, 'Perm': {'d': 14, 'gens': [482670720, 744, 1680, 3, 7192321920]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times D_{14}', 'transitive_degree': 84, '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': 4, 'aut_exponent': 84, 'aut_gen_orders': [12, 42, 12, 12], 'aut_gens': [[1, 2, 4, 112], [392, 58, 325, 225], [1, 387, 173, 112], [392, 387, 405, 449], [393, 50, 189, 616]], 'aut_group': None, 'aut_hash': 4249473474260184187, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 387072, 'aut_permdeg': 80, 'aut_perms': [56586508009837195323176692243135703299775127108038324696431473359049522850058136739924853075119318802678726375412672910, 17204376306414426621998046988490614104525944328192077488374026600541343984178571488896996617381301041739135479093991677, 6625044897811307247823909691105318904542816764043059699528308240656019930991324418332362460682380383679265701575703973, 35786134390986078877054606003329343670344161891174636431087545427573022351925245331773550100516067586023904258102969087], 'aut_phi_ratio': 2016.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 14, 4, 1], [2, 42, 4, 1], [3, 2, 1, 1], [4, 6, 4, 1], [6, 2, 1, 1], [6, 2, 6, 1], [6, 14, 8, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 2, 18, 1], [21, 4, 3, 1], [28, 6, 24, 1], [42, 4, 3, 1], [42, 4, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.C_2^4.C_7.C_3^3.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 4599629986281706675, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4.C_2^4.S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': None, 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 14, 4], [2, 42, 4], [3, 2, 1], [4, 6, 4], [6, 2, 7], [6, 14, 8], [7, 2, 3], [14, 2, 21], [21, 4, 3], [28, 6, 24], [42, 4, 21]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '84.8', 'commutator_count': 1, 'commutator_label': '42.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1172, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['168.16', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 14, 1, 4], [2, 42, 1, 4], [3, 2, 1, 1], [4, 6, 1, 4], [6, 2, 1, 7], [6, 14, 2, 4], [7, 2, 3, 1], [14, 2, 3, 7], [21, 4, 3, 1], [28, 6, 6, 4], [42, 4, 3, 7]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 155610, 'exponent': 84, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '336.219', 'hash': 1172, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [1, 2, 14, 3], 'inner_gens': [[1, 2, 4, 112], [1, 2, 108, 112], [1, 10, 4, 560], [1, 2, 228, 112]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 84, 'inner_split': True, 'inner_tex': 'S_3\\times D_7', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 68], [4, 24]], 'label': '672.1172', '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': 'D42:C2^3', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 17, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 108, 'number_divisions': 52, 'number_normal_subgroups': 148, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 472, 'number_subgroups': 3528, 'old_label': None, 'order': 672, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 231], [3, 2], [4, 24], [6, 126], [7, 6], [14, 42], [21, 12], [28, 144], [42, 84]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 1], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 4, 112], [1, 2, 4, 560], [57, 2, 4, 112], [1, 2, 4, 560], [1, 2, 60, 112], [1, 2, 4, 168], [1, 2, 340, 112], [1, 2, 4, 113], [337, 338, 340, 112], [1, 338, 340, 112], [1, 2, 68, 112], [1, 2, 4, 112]], 'outer_group': None, 'outer_hash': 1985732833063576623, 'outer_nilpotent': False, 'outer_order': 4608, 'outer_permdeg': 19, 'outer_perms': [0, 6446835113241600, 13875745172894929, 6446835113241600, 12341, 20713823524012219, 107, 6511178919181798, 6448154762661889, 13877053326259307, 4004021, 0], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6.A_4.C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 4], [6, 8], [12, 12]], 'representations': {'PC': {'code': 2332238642564394464593990503363253, 'gens': [1, 2, 3, 6], 'pres': [7, -2, -2, -2, -2, -7, -2, -3, 1143, 58, 1466, 80, 1691, 5899, 124, 5508]}, 'GLZN': {'d': 2, 'p': 56, 'gens': [7664028, 3536940, 5092893, 4741659, 7200297, 176065, 1559776]}, 'Perm': {'d': 18, 'gens': [87660926767, 21010929811201, 88698758401, 88698758400, 1520467200, 376610217984000, 50646]}}, 'schur_multiplier': [2, 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_{42}:C_2^3', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
