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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '648.723', 'ambient_counter': 723, 'ambient_order': 648, 'ambient_tex': 'C_3^3:D_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 9, 'counter': 27, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '648.723.12.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.f1.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': 12, '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': '54.12', 'subgroup_hash': 12, 'subgroup_order': 54, 'subgroup_tex': 'S_3\\times C_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '648.723', 'aut_centralizer_order': 18, 'aut_label': '12.f1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '24.14', 'aut_weyl_index': 108, 'centralizer': '72.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '6.c1.a1'], 'contains': ['24.d1.a1', '36.i1.a1', '36.j1.a1', '36.m1.a1', '36.n1.a1'], 'core': '72.a1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [7447, -1, 7999, -1, 2304, -1, 8532, -1], 'generators': [43626483, 79833843, 267361225, 79914484], 'label': '648.723.12.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.c1.a1', 'old_label': '12.f1.a1', 'projective_image': '648.723', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.f1.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': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 6, 4, 12], 'aut_gens': [[1, 3, 18], [2, 4, 18], [12, 17, 36], [14, 4, 18], [6, 46, 18]], 'aut_group': '288.851', 'aut_hash': 851, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 11, 'aut_perms': [1992360, 727201, 11753520, 5907604], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 8, 1], [3, 2, 1, 1], [3, 2, 8, 1], [6, 3, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times \\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(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, 3, 1], [3, 1, 8], [3, 2, 9], [6, 3, 8]], 'center_label': '9.2', 'center_order': 9, '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', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 2], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 4], [3, 2, 1, 1], [3, 2, 2, 4], [6, 3, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '54.12', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3], 'inner_gens': [[1, 3, 18], [1, 3, 36], [1, 39, 18]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 18], [2, 9]], 'label': '54.12', 'linC_count': 96, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 48, 'linQ_dim': 6, 'linQ_dim_count': 48, 'linR_count': 48, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*C3^2', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 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': 27, 'number_divisions': 15, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 32, 'number_subgroups': 52, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 26], [6, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 4, 18], [14, 11, 18]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [36887, 21251], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 4]], 'representations': {'PC': {'code': 469035, 'gens': [1, 2, 4], 'pres': [4, -3, -2, -3, -3, 21, 199]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [108470321895114534, 125101736062734304]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [30419024, 36826717, 23068812, 32729290]}, 'Perm': {'d': 9, 'gens': [1, 144, 45600, 3]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_3^2', 'transitive_degree': 18, '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': '4.2', '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, 2, 2, 2], 'aut_gens': [[267361225, 395452800, 131725890, 43626388, 43626483, 79914387, 79914484], [347275709, 439044004, 131725890, 79879203, 79879108, 43591204, 43591107], [267361225, 263727810, 131725890, 79879203, 43626483, 79914387, 43591107], [399087115, 263727810, 131725890, 79879108, 79879203, 79914387, 79914484], [267361225, 395452800, 131725890, 43626388, 43626483, 79914387, 43591107], [347230346, 439079283, 211604850, 79879203, 79879108, 43591204, 43591107]], 'aut_group': '2592.fs', 'aut_hash': 3641402233686599739, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2592, 'aut_permdeg': 72, 'aut_perms': [60847111713549391811551426381317158457817047345382660047437542610141509696750682343518264861634444191963, 4444542230761050037072702875762424697292592605525958829382498774012228331471827656737410279298691350114, 22282328683496954846642159379971721091921245932797786031711069926930142034116608471565915800674586229265, 1058214938124767415606612780540113714881027132716188079241698954888256802995765730760701430174840, 54791157015284993435109997748288668337915604831840117462220227777031787133904690393947873562141133546780], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 18, 1, 1], [2, 54, 1, 1], [3, 2, 1, 2], [3, 4, 1, 3], [3, 4, 2, 1], [3, 8, 1, 3], [3, 8, 2, 2], [4, 54, 1, 1], [6, 18, 1, 2], [6, 18, 2, 1], [6, 36, 1, 2], [6, 36, 2, 1], [6, 108, 1, 1], [12, 54, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3^4:C_2', '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': 12, 'autcentquo_group': '2592.fs', 'autcentquo_hash': 3641402233686599739, 'autcentquo_nilpotent': False, 'autcentquo_order': 2592, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^4:C_2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 18, 1], [2, 54, 1], [3, 2, 2], [3, 4, 5], [3, 8, 7], [4, 54, 1], [6, 18, 4], [6, 36, 4], [6, 108, 1], [12, 54, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '648.723', 'commutator_count': 1, 'commutator_label': '162.52', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 723, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 18, 1, 1], [2, 54, 1, 1], [3, 2, 1, 2], [3, 4, 1, 3], [3, 4, 2, 1], [3, 8, 1, 5], [3, 8, 2, 1], [4, 54, 1, 1], [6, 18, 1, 2], [6, 18, 2, 1], [6, 36, 1, 2], [6, 36, 2, 1], [6, 108, 1, 1], [12, 54, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 24, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 2], [8, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '648.723', 'hash': 723, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 2, 3, 3, 3, 3], 'inner_gens': [[267361225, 263727810, 131725890, 79879203, 43626483, 79914387, 43591107], [399087115, 395452800, 131725890, 43626483, 79879203, 43591204, 79914484], [267361225, 395452800, 131725890, 79879203, 79879108, 79914387, 79914484], [347230346, 439079040, 175352278, 43626388, 43626483, 79914387, 79914484], [267361225, 395453043, 175352373, 43626388, 43626483, 79914387, 79914484], [267361225, 475367187, 131725890, 43626388, 43626483, 79914387, 79914484], [303684602, 395452800, 131725890, 43626388, 43626483, 79914387, 79914484]], 'inner_hash': 723, 'inner_nilpotent': False, 'inner_order': 648, 'inner_split': True, 'inner_tex': 'C_3^3:D_{12}', 'inner_used': [1, 2, 4, 6, 7], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 9], [4, 10], [8, 7]], 'label': '648.723', 'linC_count': 16, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 26, 'linQ_dim': 8, 'linQ_dim_count': 26, 'linR_count': 42, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3:D12', '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': 24, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 30, 'number_divisions': 25, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 168, 'number_subgroup_classes': 187, 'number_subgroups': 2064, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 81], [3, 80], [4, 54], [6, 324], [12, 108]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[267361225, 395452800, 131725890, 43626388, 43626483, 79914387, 43591107], [267361225, 395452800, 131725890, 43626388, 43626483, 43591204, 79914484]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 8], [8, 7], [16, 1]], 'representations': {'PC': {'code': 225310835966370245329776073665174364943, 'gens': [1, 2, 5, 6, 7], 'pres': [7, -2, -2, -2, -3, -3, -3, 3, 309, 36, 422, 58, 451, 851, 12108, 1531, 24702, 8833, 5312]}, 'Perm': {'d': 12, 'gens': [267361225, 395452800, 131725890, 43626388, 43626483, 79914387, 79914484]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:D_{12}', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
