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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1200.923', 'ambient_counter': 923, 'ambient_order': 1200, 'ambient_tex': 'C_{10}^2.D_6', 'central': True, 'central_factor': False, 'centralizer_order': 1200, 'characteristic': True, 'core_order': 2, 'counter': 82, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1200.923.600.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '600.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '600.203', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 203, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 600, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{10}\\times D_{30}', 'simple': True, 'solvable': True, 'special_labels': ['Phi'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1200.923', 'aut_centralizer_order': 92160, 'aut_label': '600.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 92160, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['120.a1', '120.b1', '120.e1', '200.a1', '300.a1', '300.c1'], 'contains': ['1200.a1'], 'core': '600.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3056, 7557, 6097, 1910], 'generators': [20], 'label': '1200.923.600.a1', 'mobius_quo': -1, 'mobius_sub': 120, 'normal_closure': '600.a1', 'normal_contained_in': ['120.a1', '120.b1', '200.a1', '300.a1'], 'normal_contains': ['1200.a1'], 'normalizer': '1.a1', 'old_label': '600.a1', 'projective_image': '600.203', 'quotient_action_image': '1.1', 'quotient_action_kernel': '600.203', 'quotient_action_kernel_order': 600, 'quotient_fusion': None, 'short_label': '600.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], '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, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], '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': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '80.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 60, 'aut_gen_orders': [4, 4, 12, 2, 60], 'aut_gens': [[1, 2, 40], [621, 518, 280], [21, 843, 301], [600, 723, 541], [620, 1039, 1061], [621, 646, 661]], 'aut_group': None, 'aut_hash': 7196069784989913445, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 92160, 'aut_permdeg': 124, 'aut_perms': [613087717892375850752684942710355862099338357228515441659327092725294904700456059530877763118468863369906039353991907673765971614757809033250984991704816578808894366539264049030022252479643153974989023000897, 772081329358186471083211497774826572543597597844954429285799219407805580415012049317776749287259308249163482906964859472531089839405259785012429638021618660419636343459474422297359901548029458540282145612043, 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1], [30, 2, 24, 2], [30, 2, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3.(C_{30}\\times A_4).C_2^4.C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '768.1090064', 'autcent_hash': 1090064, 'autcent_nilpotent': False, 'autcent_order': 768, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times C_2^3:S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '120.36', 'autcentquo_hash': 36, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 2, 1], [4, 15, 8], [5, 1, 4], [5, 2, 10], [6, 2, 7], [10, 1, 28], [10, 2, 70], [15, 2, 24], [20, 15, 32], [30, 2, 168]], 'center_label': '40.14', 'center_order': 40, 'central_product': True, 'central_quotient': '30.3', 'commutator_count': 1, 'commutator_label': '15.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 923, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1], ['60.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 15, 2, 4], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [6, 2, 1, 7], [10, 1, 4, 7], [10, 2, 2, 7], [10, 2, 4, 14], [15, 2, 4, 2], [15, 2, 8, 2], [20, 15, 8, 4], [30, 2, 4, 14], [30, 2, 8, 14]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5208, 'exponent': 60, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '600.203', 'hash': 923, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [1, 2, 15], 'inner_gens': [[1, 2, 40], [1, 2, 1160], [1, 82, 40]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 30, 'inner_split': True, 'inner_tex': 'D_{15}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 280]], 'label': '1200.923', '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': 'C10^2.D6', '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, 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': 26, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 360, 'number_divisions': 88, 'number_normal_subgroups': 150, 'number_subgroup_autclasses': 84, 'number_subgroup_classes': 280, 'number_subgroups': 784, 'old_label': None, 'order': 1200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 120], [5, 24], [6, 14], [10, 168], [15, 48], [20, 480], [30, 336]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [4, 4, 2, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[21, 14, 440], [620, 27, 1041], [1, 622, 440], [621, 7, 1141]], 'outer_group': None, 'outer_hash': 4658869075538623801, 'outer_nilpotent': False, 'outer_order': 3072, 'outer_permdeg': 16, 'outer_perms': [1321086780857, 8034294199697, 2877362616367, 1676629353720], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:A_4.C_2^2.C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 16], [8, 36], [16, 16]], 'representations': {'PC': {'code': 19748055474384388502267198537927, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -5, -2, -3, -5, 36, 58, 20311, 102, 23532, 166, 23533]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [7187425, 287521, 6612431, 289675, 288949, 12362371, 3294907]}, 'Perm': {'d': 21, 'gens': [2446777753349897670, 289, 5329, 5250034053358617600, 288, 3991680, 7665149074781721600]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2.D_6', 'transitive_degree': 240, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [6, 30, 4, 4, 12], 'aut_gens': [[1, 2, 20], [301, 598, 380], [301, 458, 20], [1, 438, 141], [301, 134, 580], [1, 215, 261]], 'aut_group': None, 'aut_hash': 3326967402816584911, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11520, 'aut_permdeg': 64, 'aut_perms': [6011277261467391099283410270614594566127231253935879715775138114739043948271046575789306, 6011278882533298159167801140968354451112815001089518754591428582900561351514277783300635, 6011279886452392956010502207049699003114656871450798857938115738486529343197238398737591, 2045553263718459106071254640026244114277917588953394959917392604909903861101881853267257, 2045554542048660874036907828977228162701720587437272436806672216592056532554132992593320], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 15, 4, 1], [3, 2, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [6, 2, 3, 1], [10, 1, 12, 1], [10, 2, 6, 1], [10, 2, 24, 1], [10, 15, 16, 1], [15, 2, 4, 2], [15, 2, 16, 1], [30, 2, 12, 2], [30, 2, 48, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_3\\times S_4\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.186', 'autcent_hash': 186, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '120.36', 'autcentquo_hash': 36, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 15, 4], [3, 2, 1], [5, 1, 4], [5, 2, 10], [6, 2, 3], [10, 1, 12], [10, 2, 30], [10, 15, 16], [15, 2, 24], [30, 2, 72]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '30.3', 'commutator_count': 1, 'commutator_label': '15.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 203, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['30.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 15, 1, 4], [3, 2, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [6, 2, 1, 3], [10, 1, 4, 3], [10, 2, 2, 3], [10, 2, 4, 6], [10, 15, 4, 4], [15, 2, 4, 2], [15, 2, 8, 2], [30, 2, 4, 6], [30, 2, 8, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5208, 'exponent': 30, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '600.203', 'hash': 203, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [1, 2, 15], 'inner_gens': [[1, 2, 20], [1, 2, 580], [1, 42, 20]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 30, 'inner_split': True, 'inner_tex': 'D_{15}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 140]], 'label': '600.203', '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': 'C10*D30', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 180, 'number_divisions': 48, 'number_normal_subgroups': 62, 'number_subgroup_autclasses': 54, 'number_subgroup_classes': 148, 'number_subgroups': 608, 'old_label': None, 'order': 600, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 63], [3, 2], [5, 24], [6, 6], [10, 312], [15, 48], [30, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [4, 2, 4, 4, 2, 4, 6, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 15, 141], [1, 3, 221], [1, 14, 220], [1, 315, 20], [1, 18, 20], [1, 15, 20], [300, 19, 21], [1, 14, 20]], 'outer_group': '384.17862', 'outer_hash': 17862, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 12, 'outer_perms': [123758640, 123747240, 22, 87096377, 7, 127376057, 123379207, 5177], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4^2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 12], [8, 16], [16, 8]], 'representations': {'PC': {'code': 1006952610205239540293831, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -5, -2, -3, -5, 31, 6969, 69, 8410, 118, 8651]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [898450, 35962, 826574, 359380, 36301, 425755]}, 'Perm': {'d': 17, 'gens': [1313905017607, 7, 16, 46200, 43545600, 23544365644800]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}\\times D_{30}', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
