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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '432.736', 'ambient_counter': 736, 'ambient_order': 432, 'ambient_tex': 'S_3\\times F_9', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 72, 'counter': 5, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '432.736.3.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '3.a1.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': 3, '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': '144.185', 'subgroup_hash': 185, 'subgroup_order': 144, 'subgroup_tex': 'C_2\\times F_9', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '432.736', 'aut_centralizer_order': 2, 'aut_label': '3.a1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '144.182', 'aut_weyl_index': 6, 'centralizer': '216.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '6.b1.a1', '6.c1.a1', '27.a1.a1'], 'core': '6.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [484, -1, 9171, -1, 7989, -1, 8058, -1], 'generators': [1, 144, 192, 6, 12, 24], 'label': '432.736.3.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '3.a1.a1', 'projective_image': '432.736', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.a1.a1', 'subgroup_fusion': None, 'weyl_group': '72.39'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.5', '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, 8, 24], [73, 8, 24], [3, 104, 128], [17, 16, 120], [73, 112, 32], [97, 8, 24], [3, 16, 32], [137, 8, 24]], 'aut_group': '288.1027', 'aut_hash': 1027, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 11, 'aut_perms': [1, 1991070, 11492814, 2762695, 26880504, 86094, 22761055], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 8, 1, 1], [4, 9, 2, 2], [6, 8, 1, 1], [8, 9, 4, 2]], '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, 8, 1], [4, 9, 4], [6, 8, 1], [8, 9, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '72.39', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 185, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['72.39', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 8, 1, 1], [4, 9, 2, 2], [6, 8, 1, 1], [8, 9, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 24, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '144.185', 'hash': 185, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [8, 3, 3], 'inner_gens': [[1, 96, 128], [57, 8, 24], [65, 8, 24]], 'inner_hash': 39, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'F_9', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [8, 2]], 'label': '144.185', 'linC_count': 1, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*F9', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 18, 'number_divisions': 10, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 25, 'number_subgroup_classes': 27, 'number_subgroups': 138, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 19], [3, 8], [4, 36], [6, 8], [8, 72]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[73, 8, 24], [3, 8, 136]], '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': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2], [8, 2]], 'representations': {'PC': {'code': 37573582607949879250157582607557, 'gens': [1, 4, 5], 'pres': [6, 2, 2, 2, 3, 2, 3, 12, 31, 2307, 681, 111, 3844, 2050, 916, 88, 4037, 1883, 881]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8101, 10267, 13286, 15473, 7433, 13933]}, 'Perm': {'d': 11, 'gens': [420486, 1, 900174, 1315848, 4853934, 8918784]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times F_9', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [4, 3, 6, 3, 3, 2, 4, 2], 'aut_gens': [[1, 2, 48, 144], [1, 278, 144, 96], [1, 386, 48, 144], [1, 358, 48, 336], [17, 338, 48, 144], [1, 338, 48, 144], [1, 34, 96, 288], [1, 422, 240, 192], [1, 290, 96, 288]], 'aut_group': '864.4662', 'aut_hash': 4662, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 21, 'aut_perms': [41583387886209770007, 39164875057895663119, 14768965106606631913, 19312710568165929847, 17237963562867738247, 41603448397326227, 9870536685093968187, 26942032372306416255], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 9, 2, 1], [4, 27, 2, 1], [6, 18, 1, 1], [6, 24, 1, 1], [8, 9, 2, 2], [8, 27, 2, 2], [12, 18, 2, 1], [24, 18, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'F_9:D_6', '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': 24, 'autcentquo_group': '864.4662', 'autcentquo_hash': 4662, 'autcentquo_nilpotent': False, 'autcentquo_order': 864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:D_6', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [3, 2, 1], [3, 8, 1], [3, 16, 1], [4, 9, 2], [4, 27, 2], [6, 18, 1], [6, 24, 1], [8, 9, 4], [8, 27, 4], [12, 18, 2], [24, 18, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '432.736', 'commutator_count': 1, 'commutator_label': '27.5', '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': 736, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['6.1', 1], ['72.39', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 9, 2, 1], [4, 27, 2, 1], [6, 18, 1, 1], [6, 24, 1, 1], [8, 9, 4, 1], [8, 27, 4, 1], [12, 18, 2, 1], [24, 18, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 24, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '432.736', 'hash': 736, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 24, 3, 3], 'inner_gens': [[1, 34, 48, 144], [17, 2, 336, 96], [1, 146, 48, 144], [1, 194, 48, 144]], 'inner_hash': 736, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': True, 'inner_tex': 'S_3\\times F_9', 'inner_used': [1, 2, 3], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 8], [8, 2], [16, 1]], 'label': '432.736', 'linC_count': 16, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*F9', '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': 18, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 27, 'number_divisions': 15, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 58, 'number_subgroup_classes': 58, 'number_subgroups': 568, 'old_label': None, 'order': 432, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 39], [3, 26], [4, 72], [6, 42], [8, 144], [12, 36], [24, 72]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [24], 'outer_gens': [[1, 22, 336, 384]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 3], [8, 3], [16, 1]], 'representations': {'PC': {'code': 730876869095065627774429110175127461553927, 'gens': [1, 2, 6, 7], 'pres': [7, -2, -2, -2, -2, -3, -3, 3, 477, 36, 422, 58, 1123, 80, 1124, 7068, 4051, 530, 2365, 2960, 1791]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [1305911, 20256606, 22216488, 8933619, 33540338, 33542606, 29655964]}, 'Perm': {'d': 12, 'gens': [1, 7309584, 11745624, 16148160, 55243680, 88765944, 3]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times F_9', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
