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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1458.1290', 'ambient_counter': 1290, 'ambient_order': 1458, 'ambient_tex': '(C_3\\times C_9^2):C_6', 'central': False, 'central_factor': False, 'centralizer_order': 243, 'characteristic': False, 'core_order': 27, 'counter': 86, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1458.1290.54.d1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '54.d1.c1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '54.6', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 6, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 54, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_9:C_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '27.2', 'subgroup_hash': 2, 'subgroup_order': 27, 'subgroup_tex': 'C_3\\times C_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1458.1290', 'aut_centralizer_order': 4374, 'aut_label': '54.d1', 'aut_quo_index': 1, 'aut_stab_index': 3, 'aut_weyl_group': '18.5', 'aut_weyl_index': 13122, 'centralizer': '6.d1.a1', 'complements': ['27.b1.a1', '27.b1.b1', '27.b1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['18.c1.a1', '18.n1.c1', '27.k1.c1'], 'contains': ['162.c1.a1', '162.j1.c1'], 'core': '54.d1.c1', 'coset_action_label': None, 'count': 1, 'diagramx': [976, 7747, 203, 663, 9816, 7643, 9437, 670], 'generators': [6, 1098], 'label': '1458.1290.54.d1.c1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '54.d1.c1', 'normal_contained_in': ['18.c1.a1'], 'normal_contains': ['162.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '54.d1.c1', 'projective_image': '1458.1290', 'quotient_action_image': '6.2', 'quotient_action_kernel': '9.1', 'quotient_action_kernel_order': 9, 'quotient_fusion': None, 'short_label': '54.d1.c1', 'subgroup_fusion': None, 'weyl_group': '6.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '27.2', '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, 2, 2, 6], 'aut_gens': [[1, 3], [1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'aut_group': '108.28', 'aut_hash': 28, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 108, 'aut_permdeg': 11, 'aut_perms': [4556304, 11088384, 1, 169567, 26386807], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [9, 1, 18, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '108.28', 'autcent_hash': 28, 'autcent_nilpotent': False, 'autcent_order': 108, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2: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], [3, 1, 8], [9, 1, 18]], 'center_label': '27.2', 'center_order': 27, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4], [9, 1, 6, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 4, 'exponent': 9, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', '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, 27]], 'label': '27.2', 'linC_count': 216, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 9, 'linQ_dim': 8, 'linQ_dim_count': 9, 'linR_count': 54, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C9', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 27, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 8], [9, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'outer_group': '108.28', 'outer_hash': 28, 'outer_nilpotent': False, 'outer_order': 108, 'outer_permdeg': 11, 'outer_perms': [4556304, 11088384, 1, 169567, 26386807], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 3, 'primary_abelian_invariants': [3, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 3]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -3, 3, -3, 22]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [34311, 75456]}, 'Perm': {'d': 12, 'gens': [357120, 79833600, 80884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 9], '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_9', 'transitive_degree': 27, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [18, 6, 9, 6], 'aut_gens': [[1, 6, 18, 162], [715, 12, 576, 1242], [529, 120, 534, 1350], [1411, 60, 510, 216], [1039, 120, 1128, 756]], 'aut_group': None, 'aut_hash': 6418642737646769600, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 236196, 'aut_permdeg': 243, 'aut_perms': [13655027336209259016435600296638443461064678444058838414698139149385438226396533490729831832738170595714268166130414058940724394184087921128589351730430897714529926324154184581267722103046974226482441192293554535997344783900494453967571883424239068540937747701167318077968618364574422733633359578807809644679402707855901874006655751331437936739790071536080752143176453972585334494662605600579247031207773357085302968116497112836218150556579592570714284491464583281352746412708, 10078946243176964685902253398552713382091811388655003654209088041168677555651407559319877036263869638358537715806887558045666406275748926204216903530066222787558183259891057245786670805123090529383713567773024042597842690206303736206443292042237652051444379460341219798608691477176383293634955574814272241796616536941119183995868754121959203091829078739311814984704227354174724633149616636560450470962540325529135958985328106331447262266803044802895947735635918039474245213166, 17473409594416728308261561910787050575064092590755353067072590530676783914556173591794047187748058363382055296659455486272103317227228852367896574451817660813459080412479220867592149276660984139062105215347517096393318998504239952366118161485674322845451880133870334860483434998354849083230576960694010131783768361742279677461638453780498342772181204442393598703612901768985667466419328702772702463472481481501987329198579294637376837290864874171133150301668354386501089670228, 30419678851735462070300321508113604677647175598096082473591071689415756701018733866661768244312237408675316389491861041494530392326397189047297955640200331690818454664247915895903765103784192700981179537775564415576835093057519833637999885572980071528550197417236433146902268643358434264454193327823233048590985244128593985176377509645227643049069076208192420998228000186853996371259427768260161208234822228572567247545154546138431350181086769008501514379772739901499076053488], 'aut_phi_ratio': 486.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 2], [3, 2, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 27, 1, 2], [6, 243, 1, 2], [9, 6, 3, 1], [9, 6, 6, 1], [9, 6, 9, 1], [9, 6, 18, 1], [9, 54, 1, 4], [9, 54, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2.C_3^5.C_3.C_6^2', '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': 18, 'autcentquo_group': None, 'autcentquo_hash': 6418642737646769600, 'autcentquo_nilpotent': False, 'autcentquo_order': 236196, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2.C_3^5.C_3.C_6^2', 'cc_stats': [[1, 1, 1], [2, 243, 1], [3, 2, 4], [3, 6, 3], [3, 27, 2], [6, 243, 2], [9, 6, 36], [9, 54, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1458.1290', 'commutator_count': 1, 'commutator_label': '243.31', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1290, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 4], [3, 6, 1, 3], [3, 27, 2, 1], [6, 243, 2, 1], [9, 6, 1, 3], [9, 6, 3, 11], [9, 54, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6552, 'exponent': 18, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '54.13', 'hash': 1290, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 3, 9, 9], 'inner_gens': [[1, 120, 156, 810], [67, 6, 18, 162], [43, 6, 18, 162], [811, 6, 18, 162]], 'inner_hash': 1290, 'inner_nilpotent': False, 'inner_order': 1458, 'inner_split': True, 'inner_tex': '(C_3\\times C_9^2):C_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 12], [6, 39]], 'label': '1458.1290', 'linC_count': 486, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 27, 'linQ_dim': 24, 'linQ_dim_count': 27, 'linR_count': 486, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C3*C9^2):C6', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, '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': 21, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 57, 'number_divisions': 29, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 108, 'number_subgroup_classes': 186, 'number_subgroups': 4332, 'old_label': None, 'order': 1458, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 243], [3, 80], [6, 486], [9, 648]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 6, 3, 3], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 114, 84, 162], [1, 12, 36, 216], [1, 6, 990, 270], [1, 6, 18, 270]], 'outer_group': '162.52', 'outer_hash': 52, 'outer_nilpotent': False, 'outer_order': 162, 'outer_permdeg': 12, 'outer_perms': [557, 91491839, 174550320, 576], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2\\wr C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [4, 4], [6, 6], [18, 11]], 'representations': {'PC': {'code': 1578977475768692652136720795208486599251135, 'gens': [1, 3, 4, 6], 'pres': [7, -2, -3, -3, 3, -3, -3, -3, 14, 2522, 1206, 4371, 1186, 108, 3784, 34025, 23826, 166, 47634]}, 'Perm': {'d': 36, 'gens': [54487217736282327666482381192985728250, 11280948276601954921681295740699252950827, 62717, 21918875970408231070656470112386408064000, 99115, 1485194024855332787505545573236146873600, 32851444416299540001423252867761380512000]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_3\\times C_9^2):C_6', 'transitive_degree': 243, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 18, 'aut_gen_orders': [6, 9], 'aut_gens': [[1, 6], [1, 12], [7, 6]], 'aut_group': '54.6', 'aut_hash': 6, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 54, 'aut_permdeg': 9, 'aut_perms': [81640, 58746], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 3, 1, 2], [6, 9, 1, 2], [9, 6, 1, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_9:C_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': 18, 'autcentquo_group': '54.6', 'autcentquo_hash': 6, 'autcentquo_nilpotent': False, 'autcentquo_order': 54, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_9:C_6', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 1], [3, 3, 2], [6, 9, 2], [9, 6, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '54.6', 'commutator_count': 1, 'commutator_label': '9.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 3, 2, 1], [6, 9, 2, 1], [9, 6, 1, 1], [9, 6, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 18, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '18.3', 'hash': 6, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 9], 'inner_gens': [[1, 12], [49, 6]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 54, 'inner_split': True, 'inner_tex': 'C_9:C_6', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 6], [2, 3], [6, 1]], 'label': '54.6', 'linC_count': 1, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C9:C6', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 7, 'number_conjugacy_classes': 10, 'number_divisions': 7, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 13, 'number_subgroup_classes': 13, 'number_subgroups': 36, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 9], [3, 8], [6, 18], [9, 18]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, '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], [6, 1]], 'representations': {'PC': {'code': 71019161431, 'gens': [1, 3], 'pres': [4, -2, -3, -3, -3, 8, 146, 150, 46, 579]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [73199349896339667, 69483174124774144]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [16278328, 16427885, 11495158, 36731395]}, 'Perm': {'d': 9, 'gens': [5612, 148, 321840, 80884]}}, '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_9:C_6', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
