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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': 18, 'characteristic': False, 'core_order': 12, 'counter': 237, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3024.bo.84.j1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '84.j1.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': 84, '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': '36.11', 'subgroup_hash': 11, 'subgroup_order': 36, 'subgroup_tex': 'C_3\\times A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3024.bo', 'aut_centralizer_order': 36, 'aut_label': '84.j1', 'aut_quo_index': None, 'aut_stab_index': 21, 'aut_weyl_group': '24.12', 'aut_weyl_index': 756, 'centralizer': '168.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.c1.b1', '28.a1.a1', '42.i1.b1', '42.l1.b1', '42.n1.b1'], 'contains': ['252.b1.b1', '252.q1.a1', '252.s1.c1', '336.d1.b1'], 'core': '252.b1.b1', 'coset_action_label': None, 'count': 7, 'diagramx': [6038, -1, 6602, -1, 2741, -1, 7778, -1], 'generators': [1514, 192, 108, 1512], 'label': '3024.bo.84.j1.b1', 'mobius_quo': None, 'mobius_sub': 6, 'normal_closure': '12.c1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '84.j1.b1', 'projective_image': '3024.bo', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '84.j1.b1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '9.2', '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, 3, 3, 2, 2], 'aut_gens': [[1, 3, 6], [13, 3, 30], [2, 18, 15], [1, 21, 27], [25, 3, 6], [22, 3, 6], [19, 3, 6]], 'aut_group': '144.183', 'aut_hash': 183, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144, 'aut_permdeg': 7, 'aut_perms': [1, 120, 144, 3, 744, 1680], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 4, 6, 1], [6, 3, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times S_4', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 1, 2], [3, 4, 6], [6, 3, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', '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': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 4, 2, 3], [6, 3, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.11', 'hash': 11, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 21, 27], [19, 3, 6], [22, 3, 6]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 9], [3, 3]], 'label': '36.11', 'linC_count': 2, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 3, 'linQ_dim': 5, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3*A4', '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': 7, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 14, 'number_subgroups': 30, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 26], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 18, 27], [26, 18, 15]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [7, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [3, 1], [6, 1]], 'representations': {'PC': {'code': 107356617, 'gens': [1, 2, 3], 'pres': [4, -3, -2, 2, -3, 169, 326, 34]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [706461792882, 48233911639]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [16247572, 34591248, 17540971, 23068812]}, 'Perm': {'d': 7, 'gens': [147, 3, 744, 1680]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times A_4', 'transitive_degree': 12, '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': '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}
