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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '864.2912', 'ambient_counter': 2912, 'ambient_order': 864, 'ambient_tex': 'C_2^2.S_3^3', 'central': True, 'central_factor': False, 'centralizer_order': 864, 'characteristic': True, 'core_order': 2, 'counter': 113, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '864.2912.432.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '432.c1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '432.610', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 610, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 432, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2.S_3^3', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '864.2912', 'aut_centralizer_order': 82944, 'aut_label': '432.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 82944, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['144.c1', '144.h1', '144.i1', '216.a1', '216.d1'], 'contains': ['864.a1'], 'core': '432.c1', 'coset_action_label': None, 'count': 1, 'diagramx': [3908, 7249, 4075, 7292], 'generators': [72], 'label': '864.2912.432.c1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '432.c1', 'normal_contained_in': ['144.c1', '216.a1'], 'normal_contains': ['864.a1'], 'normalizer': '1.a1', 'old_label': '432.c1', 'projective_image': '432.610', 'quotient_action_image': '1.1', 'quotient_action_kernel': '432.610', 'quotient_action_kernel_order': 432, 'quotient_fusion': None, 'short_label': '432.c1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [4, 6, 6, 12], 'aut_gens': [[1, 2, 12, 144], [510, 561, 400, 720], [9, 554, 732, 144], [486, 228, 365, 440], [126, 529, 692, 144]], 'aut_group': '82944.fk', 'aut_hash': 6620745204998214566, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 82944, 'aut_permdeg': 36, 'aut_perms': [32107471080544188980471270650774639823657, 182650154014135425066044196797201833799168, 96088272306000689169269211702065541607533, 266751521773786803978079340318827151363494], 'aut_phi_ratio': 288.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 3, 1], [3, 4, 3, 1], [3, 8, 1, 1], [4, 6, 6, 1], [4, 18, 6, 1], [4, 27, 4, 1], [6, 2, 3, 3], [6, 4, 3, 3], [6, 8, 1, 3], [12, 12, 12, 2], [12, 36, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3.D_6\\wr S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': '1296.3490', 'autcentquo_hash': 3490, 'autcentquo_nilpotent': False, 'autcentquo_order': 1296, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 6, 6], [4, 18, 6], [4, 27, 4], [6, 2, 9], [6, 4, 9], [6, 8, 3], [12, 12, 24], [12, 36, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '216.162', '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', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2912, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 6, 2, 3], [4, 18, 1, 6], [4, 27, 2, 2], [6, 2, 1, 9], [6, 4, 1, 9], [6, 8, 1, 3], [12, 12, 2, 12], [12, 36, 1, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1792, 'exponent': 12, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '216.162', 'hash': 2912, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6, 3], 'inner_gens': [[1, 442, 444, 144], [509, 2, 492, 144], [433, 530, 12, 720], [1, 2, 300, 144]], 'inner_hash': 162, 'inner_nilpotent': False, 'inner_order': 216, 'inner_split': True, 'inner_tex': 'S_3^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 28], [4, 30], [8, 4]], 'label': '864.2912', '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': 'C2^2.S3^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 78, 'number_divisions': 61, 'number_normal_subgroups': 88, 'number_subgroup_autclasses': 114, 'number_subgroup_classes': 386, 'number_subgroups': 2048, 'old_label': None, 'order': 864, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 3], [3, 26], [4, 252], [6, 78], [12, 504]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 3, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 475, 0, 0, 0, 0], 'outer_gens': [[36, 654, 121, 440], [73, 82, 132, 720], [1, 442, 12, 720], [6, 60, 361, 508], [1, 2, 12, 720], [1, 10, 12, 144], [1, 74, 84, 144], [73, 74, 12, 144]], 'outer_group': '384.20163', 'outer_hash': 20163, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 12, 'outer_perms': [212340376, 167450543, 40723327, 478836840, 167328023, 167903, 127456, 90760336], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\wr S_3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 25], [8, 10]], 'representations': {'PC': {'code': 1318184212662998104857184611466568838443308163522341, 'gens': [1, 2, 4, 7], 'pres': [8, -2, -2, -3, -2, -2, -3, -2, -3, 576, 7073, 41, 1922, 450, 14211, 7883, 91, 2412, 116, 2317, 3390, 166, 3103]}, 'Perm': {'d': 21, 'gens': [122023026169551360, 377931395847736, 377938537769663, 753247913281920, 27477313920, 325, 435, 2554547108585472000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_2^2.S_3^3', 'transitive_degree': 96, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [4, 2, 2, 3, 3], 'aut_gens': [[1, 2, 12, 72], [222, 29, 260, 72], [217, 50, 60, 72], [1, 218, 12, 72], [1, 2, 156, 72], [222, 204, 77, 220]], 'aut_group': '10368.bb', 'aut_hash': 633380781765585004, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10368, 'aut_permdeg': 18, 'aut_perms': [4476211833579473, 356242689296654, 357415511760, 11863241120509, 4472812784696196], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 3, 1], [3, 2, 3, 1], [3, 4, 3, 1], [3, 8, 1, 1], [4, 18, 3, 1], [4, 27, 2, 1], [6, 2, 3, 1], [6, 4, 3, 1], [6, 8, 1, 1], [6, 12, 6, 2], [12, 36, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\wr S_3', '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': 36, 'autcentquo_group': '1296.3490', 'autcentquo_hash': 3490, 'autcentquo_nilpotent': False, 'autcentquo_order': 1296, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 3], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 18, 3], [4, 27, 2], [6, 2, 3], [6, 4, 3], [6, 8, 1], [6, 12, 12], [12, 36, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '216.162', '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', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 610, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 3], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 18, 1, 3], [4, 27, 2, 1], [6, 2, 1, 3], [6, 4, 1, 3], [6, 8, 1, 1], [6, 12, 1, 12], [12, 36, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1792, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, -1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '216.162', 'hash': 610, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6, 3], 'inner_gens': [[1, 226, 228, 72], [221, 2, 276, 72], [217, 242, 12, 360], [1, 2, 156, 72]], 'inner_hash': 162, 'inner_nilpotent': False, 'inner_order': 216, 'inner_split': True, 'inner_tex': 'S_3^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 14], [4, 15], [8, 2]], 'label': '432.610', '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': 'C2.S3^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 39, 'number_divisions': 38, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 70, 'number_subgroup_classes': 214, 'number_subgroups': 1076, 'old_label': None, 'order': 432, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 19], [3, 26], [4, 108], [6, 170], [12, 108]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [42, 0, 259, 0, 0], 'outer_gens': [[1, 44, 366, 264], [1, 2, 228, 360], [6, 12, 289, 220], [1, 2, 60, 72], [1, 2, 12, 360]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 16], [8, 2]], 'representations': {'PC': {'code': 9044686963670919677305272275105045, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -3, -2, -3, -2, -3, 3165, 36, 170, 6387, 3874, 80, 851, 1286, 124, 1203]}, 'Perm': {'d': 17, 'gens': [4284230227336, 21371697849743, 47096395320240, 67040185766400, 45360, 325, 435]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2.S_3^3', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
