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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1458.477', 'ambient_counter': 477, 'ambient_order': 1458, 'ambient_tex': 'C_{18}.C_3^4', 'central': False, 'central_factor': False, 'centralizer_order': 486, 'characteristic': False, 'core_order': 162, 'counter': 16, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1458.477.9.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '9.f1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '9.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 9, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^2', 'simple': False, 'solvable': True, 'special_labels': ['C2'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '162.26', 'subgroup_hash': 26, 'subgroup_order': 162, 'subgroup_tex': 'C_3\\times C_{54}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1458.477', 'aut_centralizer_order': 162, 'aut_label': '9.f1', 'aut_quo_index': 4, 'aut_stab_index': 36, 'aut_weyl_group': '108.24', 'aut_weyl_index': 5832, 'centralizer': '3.a1', 'complements': ['162.f1'], 'conjugacy_class_count': 36, 'contained_in': ['3.a1', '3.d1'], 'contains': ['18.f1', '27.e1', '27.h1', '27.i1'], 'core': '9.f1', 'coset_action_label': None, 'count': 36, 'diagramx': [2600, 8079, 1748, 8071], 'generators': [42515360, 533629, 3759823, 15175873, 886492], 'label': '1458.477.9.f1', 'mobius_quo': 0, 'mobius_sub': 3, 'normal_closure': '9.f1', 'normal_contained_in': ['3.a1', '3.d1'], 'normal_contains': ['18.f1', '27.e1', '27.h1', '27.i1'], 'normalizer': '1.a1', 'old_label': '9.f1', 'projective_image': '27.5', 'quotient_action_image': '1.1', 'quotient_action_kernel': '9.2', 'quotient_action_kernel_order': 9, 'quotient_fusion': None, 'short_label': '9.f1', 'subgroup_fusion': None, 'weyl_group': '3.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '162.26', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [2, 3, 3, 18], 'aut_gens': [[1, 3], [1, 159], [1, 4], [55, 58], [2, 87]], 'aut_group': '324.148', 'aut_hash': 148, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 324, 'aut_permdeg': 29, 'aut_perms': [3249967354405493934448227607, 3158029445877126345265767900984, 1617821832071853231411925936344, 328343886518461984147155259969], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [6, 1, 2, 1], [6, 1, 6, 1], [9, 1, 6, 1], [9, 1, 12, 1], [18, 1, 6, 1], [18, 1, 12, 1], [27, 1, 54, 1], [54, 1, 54, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_3\\times C_{18}):S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 18, 'autcent_group': '324.148', 'autcent_hash': 148, 'autcent_nilpotent': False, 'autcent_order': 324, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': '(C_3\\times C_{18}):S_3', '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, 1], [3, 1, 8], [6, 1, 8], [9, 1, 18], [18, 1, 18], [27, 1, 54], [54, 1, 54]], 'center_label': '162.26', 'center_order': 162, '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', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 26, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['27.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [6, 1, 2, 4], [9, 1, 6, 3], [18, 1, 6, 3], [27, 1, 18, 3], [54, 1, 18, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 36, 'exponent': 54, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.1', 'frattini_quotient': '18.5', 'hash': 26, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 162]], 'label': '162.26', 'linC_count': 5832, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 27, 'linQ_dim': 20, 'linQ_dim_count': 27, 'linR_count': 1458, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C54', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 162, 'number_divisions': 22, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 28, 'number_subgroups': 28, 'old_label': None, 'order': 162, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [3, 8], [6, 8], [9, 18], [18, 18], [27, 54], [54, 54]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 18, 'outer_gen_orders': [2, 3, 3, 18], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 159], [1, 4], [55, 58], [2, 87]], 'outer_group': '324.148', 'outer_hash': 148, 'outer_nilpotent': False, 'outer_order': 324, 'outer_permdeg': 29, 'outer_perms': [3249967354405493934448227607, 3158029445877126345265767900984, 1617821832071853231411925936344, 328343886518461984147155259969], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': '(C_3\\times C_{18}):S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 27], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [6, 6], [18, 6]], 'representations': {'PC': {'code': 14018937303, 'gens': [1, 2], 'pres': [5, -3, -2, -3, -3, -3, 26, 57, 78]}, 'GLFp': {'d': 2, 'p': 109, 'gens': [104897410, 58276368]}, 'Perm': {'d': 32, 'gens': [8222838654177922817725562880000000, 10873224326021737633873920000, 17683523987479403909087232000000, 3323121639686209319422821120, 806634631153204606767248884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 54], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_{54}', 'transitive_degree': 162, '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': '486.250', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [24, 6, 72], 'aut_gens': [[29231470, 3759823, 14880376, 28289825], [32457664, 29233657, 29231497, 28580948], [29233657, 3764170, 29233684, 13936598], [18113104, 29233657, 14882563, 14055623]], 'aut_group': None, 'aut_hash': 6182883016810243436, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 629856, 'aut_permdeg': 126, 'aut_perms': [18107518393506400730822335223089907482160627268881002374268339619526149402636521919198341277842466535692355311718971175411507383115651101436443524538335424470698009554942889978134254745789975391692013032831619678, 10597840535749727788292469617703563262708193854311063049442133718355715333996048664999369746083394475674025148283577084561244039922675614368826375650163980200799530054063511484196216773290579086656024837671575686, 16789959193512546976376050148656794030491797775847507287565650043094314799082437829234407263120120987197125265692974727626872666328618808828218989492178078031964346150973329839075301269985734312397855288429853854], 'aut_phi_ratio': 1296.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [3, 3, 24, 1], [6, 1, 2, 1], [6, 1, 6, 1], [6, 3, 24, 1], [9, 1, 6, 1], [9, 1, 12, 1], [9, 3, 48, 1], [18, 1, 6, 1], [18, 1, 12, 1], [18, 3, 48, 1], [27, 1, 54, 1], [27, 3, 144, 1], [54, 1, 54, 1], [54, 3, 144, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.(C_3\\times Q_8).C_3^4.C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 18, 'autcent_group': None, 'autcent_hash': 1941370344581000913, 'autcent_nilpotent': False, 'autcent_order': 13122, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^4.C_3^4.C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '48.29', 'autcentquo_hash': 29, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 8], [3, 3, 24], [6, 1, 8], [6, 3, 24], [9, 1, 18], [9, 3, 48], [18, 1, 18], [18, 3, 48], [27, 1, 54], [27, 3, 144], [54, 1, 54], [54, 3, 144]], 'center_label': '162.26', 'center_order': 162, 'central_product': True, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 477, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['243.50', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [3, 3, 2, 12], [6, 1, 2, 4], [6, 3, 2, 12], [9, 1, 6, 3], [9, 3, 2, 24], [18, 1, 6, 3], [18, 3, 2, 24], [27, 1, 18, 3], [27, 3, 6, 24], [54, 1, 18, 3], [54, 3, 6, 24]], 'element_repr_type': 'GLZq', 'elementary': 3, 'eulerian_function': 3790800, 'exponent': 54, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.1', 'frattini_quotient': '162.55', 'hash': 477, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 1, 1], 'inner_gens': [[29231470, 3762010, 14880376, 28289825], [29229283, 3759823, 14880376, 28289825], [29231470, 3759823, 14880376, 28289825], [29231470, 3759823, 14880376, 28289825]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': False, 'inner_tex': 'C_3^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 486], [3, 108]], 'label': '1458.477', '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': False, 'monomial': True, 'name': 'C18.C3^4', 'ngens': 7, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 594, 'number_divisions': 142, 'number_normal_subgroups': 800, 'number_subgroup_autclasses': 66, 'number_subgroup_classes': 848, 'number_subgroups': 944, 'old_label': None, 'order': 1458, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1], [3, 80], [6, 80], [9, 162], [18, 162], [27, 486], [54, 486]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 72, 'outer_gen_orders': [72, 18, 24], 'outer_gen_pows': [531442, 531442, 531442], 'outer_gens': [[18110917, 3759850, 29233684, 42757865], [29231470, 10595725, 14880376, 28406744], [24942418, 39293566, 14884750, 42934256]], 'outer_group': None, 'outer_hash': 845442520092912447, 'outer_nilpotent': False, 'outer_order': 69984, 'outer_permdeg': 36, 'outer_perms': [62642251465961070477069777239044888446589, 1207515633836101632166589549980479837247, 62884849609214564311924520305806309586417], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3:S_3.(A_4\\times \\He_3).C_6.C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 86, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 3, 9], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 80], [6, 54], [54, 6]], 'representations': {'PC': {'code': 53283262208573193335, 'gens': [1, 2, 3, 4], 'pres': [7, -3, -3, -3, -2, -3, -3, -3, 6847, 80, 137, 166]}, 'GLZq': {'d': 2, 'q': 81, 'gens': [531469, 42515360, 533629, 3759823, 712288, 14880376, 5517250]}, 'Perm': {'d': 86, 'gens': [41380175416368155988745112745799009472479429732248776287754778719360020958150638054430784024241252409770432337505308755453464, 40652359290794847034991808973511718074575851312157559633753286364251829459498548662668165429363084222876666371802049970000, 81334257004224440913308153054256842440557977479866578593871061547554296112729964987997582240783312559887627715515975202507843, 3, 121270731531317931182948841589895465213323902525033305787237769613990935983472803772698801048203240973653377888097387917724664, 161189813785295391098650346722912109089583119976423418049500792403631528793908486331254271629276598588922592655918147334441464, 281710411438055027694947944226061159480056634330574206405101912752560026159795933451040286452340924018275123200000000000000000000]}}, 'schur_multiplier': [3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 18], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{18}.C_3^4', 'transitive_degree': 486, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 3], [1, 7], [4, 3]], 'aut_group': '48.29', 'aut_hash': 29, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 8, 'aut_perms': [31834, 28334], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\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': 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], [3, 1, 8]], 'center_label': '9.2', 'center_order': 9, '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': ['3.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '9.2', 'hash': 2, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 9]], 'label': '9.2', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 6, 'linQ_dim': 4, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 9, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 9, 'order_factorization_type': 2, 'order_stats': [[1, 1], [3, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 7], [4, 3]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [31834, 28334], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 3, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -3, 3]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858733, 35931237]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [687, 1374]}, 'Perm': {'d': 6, 'gens': [240, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
