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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1728.33796', 'ambient_counter': 33796, 'ambient_order': 1728, 'ambient_tex': 'C_8:S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 36, 'counter': 475, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.33796.24.bv1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.bv1.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': 24, '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': '72.27', 'subgroup_hash': 27, 'subgroup_order': 72, 'subgroup_tex': 'S_3\\times C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.33796', 'aut_centralizer_order': 48, 'aut_label': '24.bv1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '48.51', 'aut_weyl_index': 288, 'centralizer': '36.bg1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.m1.a1', '12.y1.b1', '12.bw1.a1', '12.bz1.a1', '12.cb1.b1', '12.cc1.b1', '12.cd1.b1', '12.ce1.b1'], 'contains': ['48.f1.b1', '48.q1.b1', '48.bb1.b1', '72.bo1.a1', '72.bt1.b1'], 'core': '48.f1.b1', 'coset_action_label': None, 'count': 3, 'diagramx': [5366, -1, 4639, -1, 5233, -1, 5709, -1], 'generators': [18, 24, 432, 4, 864], 'label': '1728.33796.24.bv1.b1', 'mobius_quo': None, 'mobius_sub': 8, 'normal_closure': '8.m1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '24.bv1.b1', 'projective_image': '432.759', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.bv1.b1', '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': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 2, 2, 2, 3], 'aut_gens': [[1, 6], [37, 66], [1, 42], [41, 6], [5, 42], [25, 6]], 'aut_group': '48.51', 'aut_hash': 51, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 9, 'aut_perms': [41041, 90720, 24, 131784, 3], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 3, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 4, 1], [12, 1, 4, 1], [12, 2, 2, 1], [12, 2, 4, 1], [12, 3, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', '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, 1, 2], [3, 2, 3], [4, 1, 2], [4, 3, 2], [6, 1, 2], [6, 2, 3], [6, 3, 4], [12, 1, 4], [12, 2, 6], [12, 3, 4]], 'center_label': '12.2', 'center_order': 12, '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', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 3, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2], [12, 1, 4, 1], [12, 2, 2, 1], [12, 2, 4, 1], [12, 3, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 12, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '36.12', 'hash': 27, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 30], [49, 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': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 24], [2, 12]], 'label': '72.27', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 28, 'linQ_dim': 6, 'linQ_dim_count': 28, 'linR_count': 10, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'S3*C12', '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': 16, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 36, 'number_divisions': 18, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 31, 'number_subgroup_classes': 35, 'number_subgroups': 58, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 7], [3, 8], [4, 8], [6, 20], [12, 28]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[5, 6], [1, 42], [37, 6]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 5], [8, 1]], 'representations': {'PC': {'code': 1042831902776745, 'gens': [1, 3], 'pres': [5, -2, -3, -2, -2, -3, 10, 452, 42, 1203, 58, 1204]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101739926298686, 41624330840588829]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [182, 21974, 4396]}, 'Perm': {'d': 10, 'gens': [367921, 811440, 144, 367920, 3]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], '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_{12}', 'transitive_degree': 24, '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': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 12, 4, 6, 12, 2, 6], 'aut_gens': [[1, 2, 12, 72], [5, 620, 582, 1104], [865, 1192, 1494, 696], [9, 620, 342, 1536], [1, 332, 582, 264], [865, 58, 12, 72], [5, 332, 342, 1104], [5, 898, 1500, 792], [5, 26, 636, 1656]], 'aut_group': None, 'aut_hash': 1344521839349742178, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 13824, 'aut_permdeg': 22, 'aut_perms': [121761762435345774934, 46359009532955822846, 1119518739141284913111, 162882908216048212955, 46359007857602997422, 121761764284728101038, 162883135713249261713, 121761749377072324558], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [2, 6, 2, 1], [2, 9, 2, 1], [2, 18, 2, 1], [2, 27, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [4, 1, 2, 1], [4, 3, 2, 1], [4, 6, 2, 1], [4, 9, 2, 1], [4, 18, 2, 1], [4, 27, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 4, 2, 1], [6, 6, 4, 1], [6, 8, 1, 1], [6, 12, 2, 3], [6, 18, 2, 1], [6, 24, 2, 1], [6, 36, 2, 1], [8, 2, 2, 1], [8, 6, 2, 1], [8, 6, 4, 1], [8, 18, 2, 1], [8, 18, 4, 1], [8, 54, 2, 1], [12, 2, 2, 1], [12, 2, 4, 1], [12, 4, 2, 1], [12, 4, 4, 1], [12, 6, 4, 1], [12, 8, 2, 1], [12, 12, 2, 3], [12, 18, 2, 1], [12, 24, 2, 1], [12, 36, 2, 1], [24, 4, 2, 1], [24, 4, 4, 2], [24, 8, 4, 2], [24, 12, 4, 4], [24, 24, 4, 1], [24, 36, 2, 1], [24, 36, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3.C_2^6.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '432.741', 'autcentquo_hash': 741, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^3:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 6, 2], [2, 9, 2], [2, 18, 2], [2, 27, 2], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 1, 2], [4, 3, 2], [4, 6, 2], [4, 9, 2], [4, 18, 2], [4, 27, 2], [6, 2, 3], [6, 4, 3], [6, 6, 4], [6, 8, 1], [6, 12, 6], [6, 18, 2], [6, 24, 2], [6, 36, 2], [8, 2, 2], [8, 6, 6], [8, 18, 6], [8, 54, 2], [12, 2, 6], [12, 4, 6], [12, 6, 4], [12, 8, 2], [12, 12, 6], [12, 18, 2], [12, 24, 2], [12, 36, 2], [24, 4, 10], [24, 8, 8], [24, 12, 16], [24, 24, 4], [24, 36, 6]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '432.759', 'commutator_count': 1, 'commutator_label': '54.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 33796, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['288.439', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 9, 1, 2], [2, 18, 1, 2], [2, 27, 1, 2], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 1, 2, 1], [4, 3, 2, 1], [4, 6, 1, 2], [4, 9, 2, 1], [4, 18, 1, 2], [4, 27, 2, 1], [6, 2, 1, 3], [6, 4, 1, 3], [6, 6, 1, 4], [6, 8, 1, 1], [6, 12, 1, 6], [6, 18, 1, 2], [6, 24, 1, 2], [6, 36, 1, 2], [8, 2, 2, 1], [8, 6, 2, 3], [8, 18, 2, 3], [8, 54, 2, 1], [12, 2, 2, 3], [12, 4, 2, 3], [12, 6, 2, 2], [12, 8, 2, 1], [12, 12, 1, 4], [12, 12, 2, 1], [12, 18, 2, 1], [12, 24, 1, 2], [12, 36, 1, 2], [24, 4, 2, 3], [24, 4, 4, 1], [24, 8, 2, 2], [24, 8, 4, 1], [24, 12, 2, 6], [24, 12, 4, 1], [24, 24, 2, 2], [24, 36, 2, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 177166080, 'exponent': 24, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 4]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '432.759', 'hash': 33796, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6, 6], 'inner_gens': [[1, 10, 12, 72], [5, 2, 60, 936], [1, 26, 12, 360], [1, 866, 1452, 72]], 'inner_hash': 759, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': False, 'inner_tex': 'C_2\\times S_3^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 16, 'irrep_stats': [[1, 32], [2, 56], [4, 44], [8, 12]], 'label': '1728.33796', 'linC_count': 128, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 2432, 'linQ_dim': 10, 'linQ_dim_count': 2432, 'linR_count': 2464, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C8:S3^3', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 60, 'number_characteristic_subgroups': 74, 'number_conjugacy_classes': 144, 'number_divisions': 96, 'number_normal_subgroups': 252, 'number_subgroup_autclasses': 792, 'number_subgroup_classes': 1426, 'number_subgroups': 9940, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 127], [3, 26], [4, 128], [6, 278], [8, 256], [12, 304], [24, 608]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[865, 10, 12, 72], [1, 866, 12, 1224], [1, 2, 876, 72], [1, 10, 12, 792], [1, 44, 1158, 696]], 'outer_group': '32.51', 'outer_hash': 51, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 32], [4, 28], [8, 14], [16, 5], [32, 1]], 'representations': {'PC': {'code': 11044876816864516531877721524761089794458395039970166821969, 'gens': [1, 2, 4, 6], 'pres': [9, -2, -2, -3, -2, -3, -2, -2, -2, -3, 181, 46, 218, 1092, 102, 1093, 25286, 1652, 158, 3813, 186, 8674, 214, 7811]}, 'Perm': {'d': 17, 'gens': [5040, 20935722931343, 180587232136, 45962681990400, 68355447955200, 21109604140800, 325, 45360, 435]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8:S_3^3', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
