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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '3024.bo', 'ambient_counter': 41, 'ambient_order': 3024, 'ambient_tex': 'C_3:S_4\\times F_7', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 1, 'counter': 312, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3024.bo.168.l1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '168.l1.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': 168, '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': '18.3', 'subgroup_hash': 3, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3024.bo', 'aut_centralizer_order': 12, 'aut_label': '168.l1', 'aut_quo_index': None, 'aut_stab_index': 252, 'aut_weyl_group': '6.1', 'aut_weyl_index': 3024, 'centralizer': '504.e1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['24.l1.b1', '42.n1.b1', '56.c1.a1', '84.m1.b1'], 'contains': ['336.d1.b1', '504.o1.a1', '504.q1.b1'], 'core': '3024.a1.a1', 'coset_action_label': None, 'count': 84, 'diagramx': [8801, -1, 9324, -1, 1705, -1, 6286, -1], 'generators': [129, 1514, 192], 'label': '3024.bo.168.l1.b1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '84.m1.b1', 'old_label': '168.l1.b1', 'projective_image': '3024.bo', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '168.l1.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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 6], [5, 12], [17, 6]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': 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, 3, 1], [3, 1, 2], [3, 2, 3], [6, 3, 2]], 'center_label': '3.1', 'center_order': 3, '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', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '18.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 12], [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, 6], [2, 3]], 'label': '18.3', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*S3', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 9, 'number_subgroups': 14, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8], [6, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5, 6]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 5451, 'gens': [1, 3], 'pres': [3, -2, -3, -3, 6, 110]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780110, 20974441]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'GU'}, {'d': 2, 'q': 4, 'gens': [20, 130, 194], 'family': 'COPlus'}, {'d': 2, 'q': 2, 'gens': [20, 130, 138], 'family': 'CU'}, {'d': 1, 'q': 9, 'gens': [93882, 62619, 1930], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1374]}, 'Perm': {'d': 6, 'gens': [450, 147, 243]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times S_3', 'transitive_degree': 6, 'wreath_data': ['C_3', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', '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, 6], 'aut_gens': [[1, 6, 36, 216], [1417, 2022, 1584, 2484], [493, 2310, 1548, 2268], [2377, 1398, 1548, 1944], [481, 798, 1584, 2052]], 'aut_group': None, 'aut_hash': 7876729447257322823, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 18144, 'aut_permdeg': 32, 'aut_perms': [83261339021301480657979118187117027, 50277917381946724496412436151979728, 49745009239239775053048516095557174, 88824865388911470661585480527097483], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 7, 1, 1], [2, 18, 1, 1], [2, 21, 1, 1], [2, 126, 1, 1], [3, 2, 1, 1], [3, 7, 1, 2], [3, 8, 3, 1], [3, 14, 1, 2], [3, 56, 3, 2], [4, 18, 1, 1], [4, 126, 1, 1], [6, 6, 1, 1], [6, 7, 1, 2], [6, 14, 1, 3], [6, 21, 1, 4], [6, 42, 1, 5], [6, 56, 3, 3], [6, 126, 1, 4], [7, 6, 1, 1], [12, 126, 1, 4], [14, 18, 1, 1], [14, 108, 1, 1], [21, 12, 1, 1], [21, 48, 3, 1], [28, 108, 1, 1], [42, 36, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_7\\times C_3:S_3:S_4', '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': 84, 'autcentquo_group': None, 'autcentquo_hash': 7876729447257322823, 'autcentquo_nilpotent': False, 'autcentquo_order': 18144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_7\\times C_3:S_3:S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 7, 1], [2, 18, 1], [2, 21, 1], [2, 126, 1], [3, 2, 1], [3, 7, 2], [3, 8, 3], [3, 14, 2], [3, 56, 6], [4, 18, 1], [4, 126, 1], [6, 6, 1], [6, 7, 2], [6, 14, 3], [6, 21, 4], [6, 42, 5], [6, 56, 9], [6, 126, 4], [7, 6, 1], [12, 126, 4], [14, 18, 1], [14, 108, 1], [21, 12, 1], [21, 48, 3], [28, 108, 1], [42, 36, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '3024.bo', 'commutator_count': 1, 'commutator_label': '252.39', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 41, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['42.1', 1], ['72.43', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 7, 1, 1], [2, 18, 1, 1], [2, 21, 1, 1], [2, 126, 1, 1], [3, 2, 1, 1], [3, 7, 2, 1], [3, 8, 1, 3], [3, 14, 2, 1], [3, 56, 2, 3], [4, 18, 1, 1], [4, 126, 1, 1], [6, 6, 1, 1], [6, 7, 2, 1], [6, 14, 1, 1], [6, 14, 2, 1], [6, 21, 2, 2], [6, 42, 1, 1], [6, 42, 2, 2], [6, 56, 1, 3], [6, 56, 2, 3], [6, 126, 2, 2], [7, 6, 1, 1], [12, 126, 2, 2], [14, 18, 1, 1], [14, 108, 1, 1], [21, 12, 1, 1], [21, 48, 1, 3], [28, 108, 1, 1], [42, 36, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 524160, 'exponent': 84, 'exponents_of_order': [4, 3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[36, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '3024.bo', 'hash': 1618119482520357889, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [12, 6, 6, 14], 'inner_gens': [[1, 1542, 1692, 1944], [121, 6, 1656, 2916], [1585, 1626, 36, 216], [1297, 546, 36, 216]], 'inner_hash': 1618119482520357889, 'inner_nilpotent': False, 'inner_order': 3024, 'inner_split': True, 'inner_tex': 'C_3:S_4\\times F_7', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 36, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 36, 'irrep_stats': [[1, 12], [2, 24], [3, 12], [6, 8], [12, 4], [18, 2], [36, 1]], 'label': '3024.bo', 'linC_count': 432, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 48, 'linQ_dim': 11, 'linQ_dim_count': 48, 'linR_count': 48, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3:S4*F7', '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, 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': 49, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 63, 'number_divisions': 45, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': 304, 'number_subgroup_classes': 416, 'number_subgroups': 7736, 'old_label': None, 'order': 3024, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 175], [3, 404], [4, 144], [6, 1364], [7, 6], [12, 504], [14, 126], [21, 156], [28, 108], [42, 36]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 111], 'outer_gens': [[1513, 1650, 1548, 216], [1513, 1578, 1692, 216]], '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': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 12], [3, 4], [4, 8], [6, 8], [12, 6], [18, 2], [36, 1]], 'representations': {'PC': {'code': '5150857078907801393481452233547034588405253960603555653050348579875143290389693559417', 'gens': [1, 3, 5, 7], 'pres': [8, 2, 3, 2, 3, 2, 3, 2, 7, 16, 12105, 37010, 1378, 66, 52611, 26123, 67684, 11060, 5188, 116, 6917, 108870, 66542, 27238, 8598, 166, 55303, 55311, 27671]}, 'Perm': {'d': 14, 'gens': [12944659556, 6227786880, 12933087361, 4394880, 6231056009, 3669120, 12454041600, 11652480]}}, 'schur_multiplier': [2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3:S_4\\times F_7', 'transitive_degree': 84, 'wreath_data': None, 'wreath_product': False}
