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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '972.455', 'ambient_counter': 455, 'ambient_order': 972, 'ambient_tex': 'C_3^3:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 54, 'characteristic': False, 'core_order': 9, 'counter': 146, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '972.455.54.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '54.d1.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': 54, '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': '18.3', 'subgroup_hash': 3, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.455', 'aut_centralizer_order': 54, 'aut_label': '54.d1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '12.4', 'aut_weyl_index': 162, 'centralizer': '18.y1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['18.a1.a1', '18.c1.a1', '18.d1.a1', '18.d1.b1', '18.d1.c1', '27.d1.a1'], 'contains': ['108.a1.a1', '162.a1.a1', '162.d1.a1'], 'core': '108.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [7722, -1, 5966, -1, 6622, -1, 4456, -1], 'generators': [15477644, 11495158, 11225629], 'label': '972.455.54.d1.a1', 'mobius_quo': None, 'mobius_sub': 27, 'normal_closure': '18.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '54.d1.a1', 'projective_image': '324.121', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.d1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 6], [5, 12], [17, 6]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': 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, 2], [3, 2, 3], [6, 3, 2]], 'center_label': '3.1', 'center_order': 3, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '18.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 12], [13, 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': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 6], [2, 3]], 'label': '18.3', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*S3', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 9, 'number_subgroups': 14, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8], [6, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5, 6]], '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': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 5451, 'gens': [1, 3], 'pres': [3, -2, -3, -3, 6, 110]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780110, 20974441]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'GU'}, {'d': 2, 'q': 4, 'gens': [20, 130, 194], 'family': 'COPlus'}, {'d': 2, 'q': 2, 'gens': [20, 130, 138], 'family': 'CU'}, {'d': 1, 'q': 9, 'gens': [93882, 62619, 1930], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1374]}, 'Perm': {'d': 6, 'gens': [450, 147, 243]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times S_3', 'transitive_degree': 6, 'wreath_data': ['C_3', 'C_2', '2T1'], 'wreath_product': True}
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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': 6, 'aut_gen_orders': [3, 3, 3, 6, 3, 3, 2, 2], 'aut_gens': [[16278328, 24577758, 15477644, 24507118, 2939606, 36731395, 7046328], [39705190, 24577758, 38113169, 24507118, 2939606, 36731395, 32552094], [39705190, 24577758, 19676700, 24507118, 2939606, 36731395, 2888631], [29290330, 7138662, 15477644, 6973453, 20201582, 36731395, 7138620], [33540361, 20201582, 38113169, 20254942, 7138662, 36731395, 42803789], [29566378, 24577758, 15477644, 24507118, 2939606, 36731395, 24577806], [33540394, 24577758, 24033196, 24507118, 2939606, 36731395, 24898803], [13129048, 24577758, 15477644, 20254942, 43014227, 36731395, 7046322], [33540394, 20201582, 12564602, 20254942, 32864349, 11225629, 3158792]], 'aut_group': '1944.2342', 'aut_hash': 2342, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1944, 'aut_permdeg': 36, 'aut_perms': [137641768323005683793288867869117658908548, 18609157326579166277426273768823526021736, 341465754947625936304027935360416749859047, 241080799595870117527751128513298224709866, 21016545857909201232452484315913185729, 299671573005467741253539343434474642127414, 137801730504868448278065914730564480310233, 300122076171946317010732072993926167842424], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 2], [3, 6, 2, 1], [3, 12, 1, 2], [3, 12, 2, 1], [3, 18, 3, 1], [3, 36, 3, 1], [6, 9, 2, 1], [6, 27, 2, 2], [6, 54, 1, 3], [6, 54, 2, 2], [6, 54, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2.S_3^3', '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': 6, 'autcentquo_group': '1944.2342', 'autcentquo_hash': 2342, 'autcentquo_nilpotent': False, 'autcentquo_order': 1944, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2.S_3^3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 27, 2], [3, 1, 2], [3, 2, 3], [3, 6, 4], [3, 12, 4], [3, 18, 3], [3, 36, 3], [6, 9, 2], [6, 27, 4], [6, 54, 10]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '324.121', 'commutator_count': 1, 'commutator_label': '243.37', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 455, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 2], [3, 6, 2, 1], [3, 12, 1, 2], [3, 12, 2, 1], [3, 18, 1, 3], [3, 36, 1, 3], [6, 9, 2, 1], [6, 27, 2, 2], [6, 54, 1, 6], [6, 54, 2, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 163296, 'exponent': 6, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 4], [12, 0, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '108.39', 'hash': 455, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 2, 3, 3, 1, 3], 'inner_gens': [[16278328, 7138662, 15477644, 42578137, 2939606, 36731395, 32653911], [33540394, 24577758, 24033196, 24507118, 2939606, 36731395, 24898803], [16278328, 20201582, 15477644, 24507118, 32864349, 36731395, 20699747], [29566378, 24577758, 15477644, 24507118, 2939606, 36731395, 24577806], [16278328, 24577758, 2414724, 24507118, 2939606, 36731395, 32653863], [16278328, 24577758, 15477644, 24507118, 2939606, 36731395, 7046328], [16278370, 7138662, 32319787, 6973453, 20201582, 36731395, 7046328]], 'inner_hash': 121, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': True, 'inner_tex': 'C_3^2:S_3^2', 'inner_used': [1, 2, 3, 4, 7], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 10], [3, 8], [4, 4], [6, 10], [12, 3]], 'label': '972.455', 'linC_count': 4, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 10, 'linQ_dim': 12, 'linQ_dim_count': 10, 'linR_count': 10, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3:S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 39, 'number_divisions': 30, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 200, 'number_subgroup_classes': 304, 'number_subgroups': 3059, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 63], [3, 242], [6, 666]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [14408200, 12816793], 'outer_gens': [[16278328, 24577758, 15477644, 20254942, 2939606, 11225629, 32552094], [22167109, 24577758, 15477644, 20254942, 43014227, 36731395, 42648819]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 4], [6, 6], [12, 5], [24, 1]], 'representations': {'PC': {'code': 26623842720242294348988207191511376951, 'gens': [1, 2, 3, 4, 6, 7], 'pres': [7, 2, 3, 3, 2, 3, 3, 3, 57, 2523, 430, 80, 2524, 746, 2042, 915, 31758, 1791]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [16278328, 24577758, 15477644, 24507118, 2939606, 36731395, 7046328]}, 'Perm': {'d': 18, 'gens': [3402388038107970, 3402106385166720, 2644291928862454, 1134035330273280, 2267507354388480, 376703666887107, 711549292653507]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3^2', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
