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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1008.584', 'ambient_counter': 584, 'ambient_order': 1008, 'ambient_tex': 'C_{14}.\\PSU(3,2)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 36, 'counter': 18, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1008.584.14.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '14.c1.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': 14, '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': '72.45', 'subgroup_hash': 45, 'subgroup_order': 72, 'subgroup_tex': 'C_2\\times C_3^2:C_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1008.584', 'aut_centralizer_order': 12, 'aut_label': '14.c1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '144.187', 'aut_weyl_index': 84, 'centralizer': '504.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '7.a1.a1'], 'contains': ['28.a1.a1', '28.d1.a1', '126.c1.a1'], 'core': '28.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [2376, -1, 2339, -1, 2431, -1, 2237, -1], 'generators': [5, 336, 8, 2, 16], 'label': '1008.584.14.c1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '14.c1.a1', 'projective_image': '504.171', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '14.c1.a1', 'subgroup_fusion': None, 'weyl_group': '72.41'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 4, 3, 2, 6], 'aut_gens': [[1, 4, 12], [37, 4, 12], [3, 32, 20], [33, 8, 60], [37, 56, 16], [5, 4, 12], [3, 4, 68], [65, 4, 12]], 'aut_group': '288.1027', 'aut_hash': 1027, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 11, 'aut_perms': [1, 2648166, 31341990, 2057041, 24356430, 2032800, 4843489], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 4, 2, 1], [4, 9, 4, 1], [6, 4, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_9:C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '144.182', 'autcentquo_hash': 182, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 4, 2], [4, 9, 4], [6, 4, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '36.9', 'commutator_count': 1, 'commutator_label': '9.2', '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': 45, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['36.9', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 4, 1, 2], [4, 9, 2, 2], [6, 4, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 12, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '72.45', 'hash': 45, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 3, 3], 'inner_gens': [[1, 56, 16], [33, 4, 12], [9, 4, 12]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': False, 'inner_tex': 'C_3^2:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [4, 4]], 'label': '72.45', 'linC_count': 2, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 2, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C3^2:C4', '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': 7, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 12, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 19, 'number_subgroup_classes': 26, 'number_subgroups': 108, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 19], [3, 8], [4, 36], [6, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [1, 0, 0], 'outer_gens': [[1, 32, 44], [37, 4, 12], [3, 8, 16]], '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': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 4]], 'representations': {'PC': {'code': 5411268416718830885, 'gens': [1, 3, 4], 'pres': [5, 2, 2, 3, 2, 3, 10, 842, 67, 323, 608, 58, 804, 609]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20468524, 31266610]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [16990, 8101, 13286, 15473, 13933]}, 'Perm': {'d': 8, 'gens': [5286, 1, 414, 11934, 2280]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_3^2:C_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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [4, 6, 12, 6, 12, 42], 'aut_gens': [[1, 4, 16, 48], [363, 316, 688, 992], [379, 142, 32, 96], [355, 958, 672, 448], [715, 772, 16, 624], [3, 292, 336, 464], [705, 638, 16, 48]], 'aut_group': None, 'aut_hash': 8254711973562652166, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12096, 'aut_permdeg': 288, 'aut_perms': 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558019203562469813628303175351500232975617212931702301525560363751787475626877455686391498734976732174215278100762498349090967601076756008959193541107639784742888534262169800670159545167410596220137953826094574369809062666651988503940774844715155708883012264990034563153087096795602120363612316081001071681604527183424875528934498229631170194523882054858010831132535683412502851373884317310501273566466197202242743085068677654441236040719270466076708440922184415470913718044540346193926772166327763071844429920994356742366095020930298085286154008395988386150772141846832914890040459220], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 8, 1, 1], [4, 18, 2, 1], [4, 126, 2, 2], [6, 8, 1, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 18, 3, 2], [21, 8, 6, 1], [28, 18, 12, 1], [42, 8, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\PSU(3,2)\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': None, 'autcentquo_hash': 2606904743955648058, 'autcentquo_nilpotent': False, 'autcentquo_order': 3024, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PSU(3,2)\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 8, 1], [4, 18, 2], [4, 126, 4], [6, 8, 1], [7, 2, 3], [14, 2, 3], [14, 18, 6], [21, 8, 6], [28, 18, 12], [42, 8, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '504.171', 'commutator_count': 1, 'commutator_label': '126.14', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 584, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 8, 1, 1], [4, 18, 2, 1], [4, 126, 1, 2], [4, 126, 2, 1], [6, 8, 1, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 18, 3, 2], [21, 8, 6, 1], [28, 18, 12, 1], [42, 8, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 84, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 0, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '504.171', 'hash': 584, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [4, 4, 3, 21], 'inner_gens': [[1, 12, 688, 416], [9, 4, 336, 320], [337, 692, 16, 48], [689, 788, 16, 48]], 'inner_hash': 171, 'inner_nilpotent': False, 'inner_order': 504, 'inner_split': True, 'inner_tex': 'C_7:\\PSU(3,2)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 26], [8, 14]], 'label': '1008.584', 'linC_count': 6, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 2, 'linQ_dim': 14, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C14.PSU(3,2)', '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': 16, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 48, 'number_divisions': 17, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 58, 'number_subgroup_classes': 62, 'number_subgroups': 776, 'old_label': None, 'order': 1008, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 19], [3, 8], [4, 540], [6, 8], [7, 6], [14, 114], [21, 48], [28, 216], [42, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [2, 2, 2], 'outer_gens': [[3, 4, 336, 752], [9, 4, 672, 736], [9, 6, 704, 512]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [40320, 24, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [6, 2], [8, 2], [12, 3], [48, 2]], 'representations': {'PC': {'code': 372942535836290790173283938196169992599237659052975, 'gens': [1, 3, 5, 6], 'pres': [7, -2, -2, -2, -2, -3, 3, -7, 14, 254, 58, 24084, 571, 2958, 165, 17477, 8076, 3379, 2042, 166, 10604]}, 'Perm': {'d': 20, 'gens': [7201450402236845, 13919073262617607, 20734746792883200, 16, 141647555303654400, 257584476991564800, 4364760]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{14}.\\PSU(3,2)', 'transitive_degree': 168, 'wreath_data': None, 'wreath_product': False}
