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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.295', 'ambient_counter': 295, 'ambient_order': 960, 'ambient_tex': 'C_{60}.C_4^2', 'central': False, 'central_factor': False, 'centralizer_order': 480, 'characteristic': False, 'core_order': 120, 'counter': 17, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.295.4.i1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.i1.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': 4, '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': '240.81', 'subgroup_hash': 81, 'subgroup_order': 240, 'subgroup_tex': 'C_4\\times C_{60}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.295', 'aut_centralizer_order': 40, 'aut_label': '4.i1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '128.2154', 'aut_weyl_index': 80, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.f1.a1', '8.j1.a1', '8.j1.b1', '12.i1.a1', '20.i1.a1'], 'core': '8.f1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [7028, -1, 2529, -1, 7775, -1, 2811, -1], 'generators': [722, 480, 320, 16, 20, 40], 'label': '960.295.4.i1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '4.i1.a1', 'projective_image': '80.14', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.i1.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': '240.81', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 4, 12, 2, 2], 'aut_gens': [[1, 4], [3, 44], [121, 126], [60, 17], [61, 91], [3, 126], [121, 6]], 'aut_group': '768.1088672', 'aut_hash': 1088672, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 14, 'aut_perms': [6267300480, 5160, 19678997520, 19766085137, 6227383680, 6266937600], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [4, 1, 12, 1], [5, 1, 4, 1], [6, 1, 6, 1], [10, 1, 12, 1], [12, 1, 24, 1], [15, 1, 8, 1], [20, 1, 48, 1], [30, 1, 24, 1], [60, 1, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6.D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '768.1088672', 'autcent_hash': 1088672, 'autcent_nilpotent': False, 'autcent_order': 768, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^6.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], [4, 1, 12], [5, 1, 4], [6, 1, 6], [10, 1, 12], [12, 1, 24], [15, 1, 8], [20, 1, 48], [30, 1, 24], [60, 1, 96]], 'center_label': '240.81', 'center_order': 240, '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', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 81, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 1, 2, 6], [5, 1, 4, 1], [6, 1, 2, 3], [10, 1, 4, 3], [12, 1, 4, 6], [15, 1, 8, 1], [20, 1, 8, 6], [30, 1, 8, 3], [60, 1, 16, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 60, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '60.13', 'hash': 81, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 4], [1, 4]], '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, 240]], 'label': '240.81', 'linC_count': 9216, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 288, 'linQ_dim': 10, 'linQ_dim_count': 288, 'linR_count': 2304, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4*C60', 'ngens': 6, 'nilpotency_class': 1, 'nilpotent': True, '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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 240, 'number_divisions': 40, 'number_normal_subgroups': 60, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 60, 'number_subgroups': 60, 'old_label': None, 'order': 240, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 12], [5, 4], [6, 6], [10, 12], [12, 24], [15, 8], [20, 48], [30, 24], [60, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 4, 12, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 44], [121, 126], [60, 17], [61, 91], [3, 126], [121, 6]], 'outer_group': '768.1088672', 'outer_hash': 1088672, 'outer_nilpotent': False, 'outer_order': 768, 'outer_permdeg': 14, 'outer_perms': [6267300480, 5160, 19678997520, 19766085137, 6227383680, 6266937600], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6.D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 10], [8, 10], [16, 6]], 'representations': {'PC': {'code': 60626298061574151, 'gens': [1, 3], 'pres': [6, -2, -2, -2, -2, -3, -5, 12, 50, 69, 118]}, 'GLFp': {'d': 2, 'p': 61, 'gens': [11349061, 10441134]}, 'Perm': {'d': 16, 'gens': [4097379686400, 127008000, 10080, 96, 1313901388800, 40279680]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 60], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times C_{60}', 'transitive_degree': 240, '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': '48.20', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 10, 2, 4, 10, 4, 4, 4, 4], 'aut_gens': [[1, 4, 80], [73, 510, 560], [19, 526, 880], [515, 556, 560], [59, 270, 80], [777, 44, 880], [793, 14, 560], [723, 766, 80], [723, 276, 880], [257, 254, 400]], 'aut_group': None, 'aut_hash': 2074058729919018285, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10240, 'aut_permdeg': 98, 'aut_perms': [7951132313322600807316628442355828254775397261387224704368180960733459049580612757723415095744070116892632540632040682169301779399706257571357861880733024, 6983153183743253026263231939308643605473711228033389427919537097744939740137492748797852526874117214416774833134549739041660116041166344621074384381739524, 5359734968630412334586988809582165408808922130480960666157785198448364752076299975307570110922502422080490121918038251498863089618968703433804176352389063, 4622743940399351862101478359129343056937565525335479226197230184160319769414497045516236824282351228528750277406237261548223242551382102817349165736214138, 5268346942138878511838307591518458351062002883873835966153431019797731391434275890174072734445171683814131861952945644517163547814611553641842624286201458, 4775256494054427529361639901259453334796511360697074571353953140049935758836292555796845943411729222812448947683630912326465295034632693419067472814639727, 7614882574476155224661572866265896095453078262031988948580454129821798613524271870275295497389150547137386409306939922212712946389075306002882962579634182, 7751897479238837560409677407180725062704371268615900559496688948967499163607072984547676662446103291734642702975816520424033592400379612326592908302401638, 5680997949247108439111266509031940655598751940986927128002947794757064049285243025559718171628031903829481135619153298524311754494781834007777862525765136], 'aut_phi_ratio': 40.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2], [3, 1, 2, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 8, 1], [4, 20, 4, 1], [5, 2, 2, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 2, 2, 2], [8, 20, 4, 1], [10, 2, 2, 1], [10, 2, 4, 3], [12, 1, 4, 2], [12, 2, 2, 2], [12, 2, 16, 1], [12, 20, 8, 1], [15, 2, 4, 1], [20, 2, 4, 4], [20, 2, 32, 1], [24, 20, 8, 1], [30, 2, 4, 1], [30, 2, 8, 3], [60, 2, 8, 4], [60, 2, 64, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_2^4.C_2^6.C_2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.2163', 'autcent_hash': 2163, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4:D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '80.50', 'autcentquo_hash': 50, 'autcentquo_nilpotent': False, 'autcentquo_order': 80, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [3, 1, 2], [4, 1, 4], [4, 2, 10], [4, 20, 4], [5, 2, 2], [6, 1, 6], [6, 2, 4], [8, 20, 4], [10, 2, 14], [12, 1, 8], [12, 2, 20], [12, 20, 8], [15, 2, 4], [20, 2, 48], [24, 20, 8], [30, 2, 28], [60, 2, 96]], 'center_label': '24.9', 'center_order': 24, 'central_product': True, 'central_quotient': '40.6', 'commutator_count': 1, 'commutator_label': '20.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 295, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['320.81', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [3, 1, 2, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 4], [4, 20, 2, 2], [5, 2, 2, 1], [6, 1, 2, 3], [6, 2, 2, 2], [8, 20, 2, 2], [10, 2, 2, 3], [10, 2, 4, 2], [12, 1, 4, 2], [12, 2, 2, 2], [12, 2, 4, 4], [12, 20, 4, 2], [15, 2, 4, 1], [20, 2, 4, 4], [20, 2, 8, 4], [24, 20, 4, 2], [30, 2, 4, 3], [30, 2, 8, 2], [60, 2, 8, 4], [60, 2, 16, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '60.10', 'hash': 295, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [2, 20, 1], 'inner_gens': [[1, 316, 80], [729, 4, 80], [1, 4, 80]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': True, 'inner_tex': 'D_{20}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [2, 228]], 'label': '960.295', 'linC_count': 2048, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 32, 'linQ_dim': 12, 'linQ_dim_count': 216, 'linR_count': 544, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C60.C4^2', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 44, 'number_characteristic_subgroups': 78, 'number_conjugacy_classes': 276, 'number_divisions': 64, 'number_normal_subgroups': 102, 'number_subgroup_autclasses': 160, 'number_subgroup_classes': 220, 'number_subgroups': 588, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 104], [5, 4], [6, 14], [8, 80], [10, 28], [12, 208], [15, 8], [20, 96], [24, 160], [30, 56], [60, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 724, 560], [1, 284, 400], [3, 724, 880], [1, 526, 880], [3, 788, 880], [721, 766, 400]], 'outer_group': '256.55630', 'outer_hash': 55630, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 14, 'outer_perms': [288, 126, 6267305809, 39917214, 6268107295, 19639065727], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 14], [4, 18], [8, 14], [16, 10], [32, 4]], 'representations': {'PC': {'code': 61007805223556171953611780575502074049, 'gens': [1, 3, 6], 'pres': [8, -2, -2, -2, -2, -5, -2, -2, -3, 16, 7586, 66, 17667, 91, 2564, 141, 166]}, 'GLZN': {'d': 2, 'p': 65, 'gens': [275471, 14005926, 4394016, 16882151, 3844764, 5767146, 12316234, 7417513]}, 'Perm': {'d': 20, 'gens': [128403161582941195, 12329, 1394852659200, 256094948229120000, 3043, 17824, 13050, 559198080]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{60}.C_4^2', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
