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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '720.659', 'ambient_counter': 659, 'ambient_order': 720, 'ambient_tex': 'C_5\\times D_6:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 60, 'counter': 33, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '720.659.6.k1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.k1.a1', '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': 6, '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': '120.22', 'subgroup_hash': 22, 'subgroup_order': 120, 'subgroup_tex': 'S_3\\times C_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '720.659', 'aut_centralizer_order': 4, 'aut_label': '6.k1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '96.206', 'aut_weyl_index': 12, 'centralizer': '36.f1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.e1.a1', '3.b1.a1'], 'contains': ['12.c1.a1', '12.o1.a1', '12.p1.a1', '18.e1.a1', '30.k1.a1'], 'core': '12.c1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [8525, -1, 8247, -1, 7237, -1, 7858, -1], 'generators': [5, 144, 360, 2, 480], 'label': '720.659.6.k1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.e1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '6.k1.a1', 'projective_image': '72.46', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.k1.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 4, 2, 6], 'aut_gens': [[1, 2], [1, 82], [61, 74], [61, 2], [21, 62]], 'aut_group': '96.206', 'aut_hash': 206, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 11, 'aut_perms': [363007, 294, 5041, 3991807], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 3, 2, 1], [5, 1, 4, 1], [6, 2, 1, 1], [10, 1, 4, 1], [10, 3, 8, 1], [12, 2, 2, 1], [15, 2, 4, 1], [20, 1, 8, 1], [20, 3, 8, 1], [30, 2, 4, 1], [60, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{12}:C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', '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, 1, 1], [2, 3, 2], [3, 2, 1], [4, 1, 2], [4, 3, 2], [5, 1, 4], [6, 2, 1], [10, 1, 4], [10, 3, 8], [12, 2, 2], [15, 2, 4], [20, 1, 8], [20, 3, 8], [30, 2, 4], [60, 2, 8]], 'center_label': '20.2', 'center_order': 20, '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', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 22, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [4, 1, 2, 1], [4, 3, 2, 1], [5, 1, 4, 1], [6, 2, 1, 1], [10, 1, 4, 1], [10, 3, 4, 2], [12, 2, 2, 1], [15, 2, 4, 1], [20, 1, 8, 1], [20, 3, 8, 1], [30, 2, 4, 1], [60, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 60, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[2, 0, 8]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '60.11', 'hash': 22, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 82], [41, 2]], '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': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 40], [2, 20]], 'label': '120.22', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 20, 'linQ_dim': 8, 'linQ_dim_count': 20, 'linR_count': 20, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'S3*C20', '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': 16, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 60, 'number_divisions': 18, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 32, 'number_subgroups': 52, 'old_label': None, 'order': 120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 8], [5, 4], [6, 2], [10, 28], [12, 4], [15, 8], [20, 32], [30, 8], [60, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[61, 2], [1, 62], [1, 74]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [8, 4], [16, 1]], 'representations': {'PC': {'code': 3378266908120971271, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -3, -5, 821, 26, 662, 42, 1763, 78]}, 'GLFp': {'d': 2, 'p': 41, 'gens': [2756841, 94465]}, 'Perm': {'d': 12, 'gens': [745, 1704, 43948800, 744, 3]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_{20}', 'transitive_degree': 60, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 4, 2, 4, 12, 4], 'aut_gens': [[1, 2, 4, 24], [377, 2, 380, 456], [377, 602, 20, 408], [9, 482, 380, 264], [17, 602, 20, 408], [377, 482, 4, 168], [1, 242, 364, 408]], 'aut_group': '1152.153163', 'aut_hash': 8169053273075801794, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1152, 'aut_permdeg': 18, 'aut_perms': [1621260408220314, 203847142749074, 1635205571531435, 203759957240592, 120156008513810, 1625789170536389], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 2, 1], [2, 6, 1, 2], [2, 18, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 18, 1, 1], [5, 1, 4, 1], [6, 2, 1, 2], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 6, 2, 2], [6, 12, 1, 1], [10, 1, 4, 1], [10, 2, 4, 1], [10, 3, 8, 1], [10, 6, 4, 2], [10, 18, 4, 1], [12, 12, 1, 1], [15, 2, 4, 2], [15, 4, 4, 1], [20, 6, 4, 1], [20, 18, 4, 1], [30, 2, 4, 2], [30, 2, 8, 1], [30, 4, 4, 2], [30, 4, 8, 1], [30, 6, 8, 2], [30, 12, 4, 1], [60, 12, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6^2.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 2], [2, 6, 2], [2, 18, 1], [3, 2, 2], [3, 4, 1], [4, 6, 1], [4, 18, 1], [5, 1, 4], [6, 2, 4], [6, 4, 4], [6, 6, 4], [6, 12, 1], [10, 1, 4], [10, 2, 4], [10, 3, 8], [10, 6, 8], [10, 18, 4], [12, 12, 1], [15, 2, 8], [15, 4, 4], [20, 6, 4], [20, 18, 4], [30, 2, 16], [30, 4, 16], [30, 6, 16], [30, 12, 4], [60, 12, 4]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '72.46', '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', '3.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 659, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.8', 1], ['5.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 18, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 6, 1, 1], [4, 18, 1, 1], [5, 1, 4, 1], [6, 2, 1, 2], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 6, 1, 2], [6, 6, 2, 1], [6, 12, 1, 1], [10, 1, 4, 1], [10, 2, 4, 1], [10, 3, 4, 2], [10, 6, 4, 2], [10, 18, 4, 1], [12, 12, 1, 1], [15, 2, 4, 2], [15, 4, 4, 1], [20, 6, 4, 1], [20, 18, 4, 1], [30, 2, 4, 2], [30, 2, 8, 1], [30, 4, 4, 2], [30, 4, 8, 1], [30, 6, 4, 2], [30, 6, 8, 1], [30, 12, 4, 1], [60, 12, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 83328, 'exponent': 60, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 8]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '360.153', 'hash': 659, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6, 3], 'inner_gens': [[1, 2, 380, 24], [1, 2, 4, 264], [369, 2, 4, 24], [1, 482, 4, 24]], 'inner_hash': 46, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'S_3\\times D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 40], [2, 70], [4, 25]], 'label': '720.659', 'linC_count': 392, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 384, 'linQ_dim': 10, 'linQ_dim_count': 384, 'linR_count': 32, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*D6:D6', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 44, 'number_characteristic_subgroups': 64, 'number_conjugacy_classes': 135, 'number_divisions': 48, 'number_normal_subgroups': 72, 'number_subgroup_autclasses': 208, 'number_subgroup_classes': 232, 'number_subgroups': 824, 'old_label': None, 'order': 720, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 39], [3, 8], [4, 24], [5, 4], [6, 60], [10, 156], [12, 12], [15, 32], [20, 96], [30, 240], [60, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 362, 20, 264], [1, 2, 364, 24], [1, 2, 4, 552]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 16, 'outer_perms': [1321087219489, 2878331623996, 4298736177138], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 13], [8, 11], [16, 5], [32, 1]], 'representations': {'PC': {'code': 2455537247251935316495653732423, 'gens': [1, 2, 3, 5], 'pres': [7, -2, -2, -2, -3, -2, -3, -5, 7982, 58, 451, 4631, 102, 11100, 166]}, 'GLZN': {'d': 2, 'p': 44, 'gens': [1530963, 2347879, 1959255, 766665, 2445403, 1044979, 86659]}, 'Perm': {'d': 15, 'gens': [6227020800, 745, 264, 43948800, 1680, 3, 93405312000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_6:D_6', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
