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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '222336.a', 'ambient_counter': 1, 'ambient_order': 222336, 'ambient_tex': 'C_6\\times F_{193}', 'central': False, 'central_factor': False, 'centralizer_order': 1152, 'characteristic': False, 'core_order': 6, 'counter': 202, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '222336.a.18528.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '18528.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': 18528, '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': '12.5', 'subgroup_hash': 5, 'subgroup_order': 12, 'subgroup_tex': 'C_2\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '222336.a', 'aut_centralizer_order': None, 'aut_label': '18528.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '193.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['96.a1.a1', '6176.a1.a1', '9264.a1.a1'], 'contains': ['37056.a1.a1', '37056.c1.a1', '37056.e1.a1', '55584.a1.a1'], 'core': '37056.a1.a1', 'coset_action_label': None, 'count': 193, 'diagramx': [1189, -1, 2033, -1, 2661, -1, 2718, -1], 'generators': [96, 111168, 74112], 'label': '222336.a.18528.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '96.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '193.a1.a1', 'old_label': '18528.a1.a1', 'projective_image': '37056.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18528.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '12.5', '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, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', '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, 3], [3, 1, 2], [6, 1, 6]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_6', 'transitive_degree': 12, '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': '1152.2487', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 37056, 'aut_gen_orders': [24, 192, 96, 64], 'aut_gens': [[1, 192], [108481, 211776], [164353, 155712], [38401, 40512], [66241, 129216]], 'aut_group': None, 'aut_hash': 5408501734751148099, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 444672, 'aut_permdeg': 965, 'aut_perms': 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'aut_phi_ratio': 6.03125, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 1, 2], [3, 1, 2, 1], [3, 193, 3, 2], [4, 193, 1, 4], [6, 1, 2, 1], [6, 193, 2, 2], [6, 193, 3, 6], [8, 193, 1, 8], [12, 193, 2, 4], [12, 193, 3, 8], [16, 193, 1, 16], [24, 193, 2, 8], [24, 193, 3, 16], [32, 193, 1, 32], [48, 193, 2, 16], [48, 193, 3, 32], [64, 193, 2, 32], [96, 193, 2, 32], [96, 193, 3, 64], [192, 193, 4, 32], [192, 193, 6, 64], [193, 192, 1, 1], [386, 192, 1, 1], [579, 192, 2, 1], [1158, 192, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{579}.C_{96}.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 37056, 'autcentquo_group': '37056.a', 'autcentquo_hash': 1432757702296672986, 'autcentquo_nilpotent': False, 'autcentquo_order': 37056, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{193}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 193, 2], [3, 1, 2], [3, 193, 6], [4, 193, 4], [6, 1, 2], [6, 193, 22], [8, 193, 8], [12, 193, 32], [16, 193, 16], [24, 193, 64], [32, 193, 32], [48, 193, 128], [64, 193, 64], [96, 193, 256], [192, 193, 512], [193, 192, 1], [386, 192, 1], [579, 192, 2], [1158, 192, 2]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '37056.a', 'commutator_count': 1, 'commutator_label': '193.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '193.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['37056.a', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 193, 1, 2], [3, 1, 2, 1], [3, 193, 2, 3], [4, 193, 2, 2], [6, 1, 2, 1], [6, 193, 2, 11], [8, 193, 4, 2], [12, 193, 4, 8], [16, 193, 8, 2], [24, 193, 8, 8], [32, 193, 16, 2], [48, 193, 16, 8], [64, 193, 32, 2], [96, 193, 32, 8], [192, 193, 64, 8], [193, 192, 1, 1], [386, 192, 1, 1], [579, 192, 2, 1], [1158, 192, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24576, 'exponent': 37056, 'exponents_of_order': [7, 2, 1], 'factors_of_aut_order': [2, 3, 193], 'factors_of_order': [2, 3, 193], 'faithful_reps': [[192, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '222336.a', 'hash': 6561991287343693459, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 37056, 'inner_gen_orders': [192, 193], 'inner_gens': [[1, 124608], [97921, 192]], 'inner_hash': 1432757702296672986, 'inner_nilpotent': False, 'inner_order': 37056, 'inner_split': True, 'inner_tex': 'F_{193}', 'inner_used': [1, 2], 'irrC_degree': 192, 'irrQ_degree': 384, 'irrQ_dim': 384, 'irrR_degree': None, 'irrep_stats': [[1, 1152], [192, 6]], 'label': '222336.a', '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': True, 'monomial': True, 'name': 'C6*F193', 'ngens': 10, '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, 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': 388, 'number_characteristic_subgroups': 58, 'number_conjugacy_classes': 1158, 'number_divisions': 74, 'number_normal_subgroups': 124, 'number_subgroup_autclasses': 152, 'number_subgroup_classes': 240, 'number_subgroups': 22512, 'old_label': None, 'order': 222336, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 387], [3, 1160], [4, 772], [6, 4248], [8, 1544], [12, 6176], [16, 3088], [24, 12352], [32, 6176], [48, 24704], [64, 12352], [96, 49408], [192, 98816], [193, 192], [386, 192], [579, 384], [1158, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[74113, 222144], [185281, 73920]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [7, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 198, 'pgroup': 0, 'primary_abelian_invariants': [2, 64, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 10], [8, 10], [16, 10], [32, 10], [64, 8], [192, 2], [384, 2]], 'representations': {'PC': {'code': '52964682967641377907896681421989232434088034869040398266885333928732307941588737271089122124336872206934025231728433068783575157474602557766028293402790237376935292542585413924364227151921535', 'gens': [1, 8], 'pres': [10, -2, -2, -2, -2, -2, -2, -3, -2, -3, -193, 20, 51, 82, 113, 144, 175, 9968647, 6504977, 2330907, 393637, 323567, 40857, 66547, 237, 2419208, 4631058, 241948, 885638, 727968, 91858, 149648, 358, 8064009, 4320019, 806429, 172839, 1036849, 306059, 151269]}, 'GLFp': {'d': 2, 'p': 193, 'gens': [201293672, 7189251, 783607298]}, 'Perm': {'d': 198, 'gens': [1047421830653597612330533414845238952812505590283855614739491814281166297074443161463084420437179133995575520943526372931497001630109407254277964699720193413017851813801636178292071115477943936670968119175441868907875334170944206242052743702773266140415258461787406913100595941039423958898035672046959747792159495252288510684858634378209668449377331496686605990344881, 101602750195964953953589546617674807098570923018844520454018572359391413496595760501485959185647492005787901467365035814473868885883969016951897395135516391823791123268512470252303362965534608128875126838651703866143143510203223691523034117338325794230421471670336666883357623062316528591914662758619489007795945935644871400384451352319863482567693348185301812684670720, 2066064867180288019123252549211314056623246063700464255459928418729304120129156917935457987192712589879421325717125246713763548873333132049412169845567610642895569482029121183903659061279779709976786975676333219215836761257091673963105940301738853538857155195182072852785454064254990979270642536313778406870556765145945886859366559256060276776488223646114535807128271]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 192], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times F_{193}', 'transitive_degree': 1158, 'wreath_data': None, 'wreath_product': False}
