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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1944.3755', 'ambient_counter': 3755, 'ambient_order': 1944, 'ambient_tex': 'C_2\\times C_3^3:C_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 72, 'characteristic': False, 'core_order': 1, 'counter': 284, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1944.3755.972.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '972.b1', 'outer_equivalence': True, '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': 972, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, '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': '1944.3755', 'aut_centralizer_order': 7776, 'aut_label': '972.b1', 'aut_quo_index': None, 'aut_stab_index': 108, 'aut_weyl_group': '1.1', 'aut_weyl_index': 839808, 'centralizer': '27.a1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['324.k1', '324.l1', '324.m1', '324.n1', '324.o1', '486.b1'], 'contains': ['1944.a1'], 'core': '1944.a1', 'coset_action_label': None, 'count': 108, 'diagramx': [2102, -1, 1606, -1], 'generators': [47159], 'label': '1944.3755.972.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '36.g1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.a1', 'old_label': '972.b1', 'projective_image': '1944.3755', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '972.b1', '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': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '72.50', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 6, 6, 6, 12, 6], 'aut_gens': [[46715, 1166425, 78793, 1668518, 793601], [1380691, 606541, 486277, 1668518, 793601], [836503, 606541, 89494, 922006, 794033], [276211, 606541, 89494, 1225079, 794033], [548929, 606541, 1591453, 1668086, 886915], [264439, 1166425, 1182673, 805166, 1633427], [89435, 606541, 90358, 1225079, 886915]], 'aut_group': None, 'aut_hash': 4687953029827479755, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 839808, 'aut_permdeg': 126, 'aut_perms': [1662431110393947826455945072365509146020862377704839136966633465900383655615652089324476684313438868023102661946964552727039971262050921012706927631334830657462272418678363953269175303286625128522323785283111151, 2972260134856278098109653702672207912693059373475393138759023119411016911907003166210284530441362242043512439826051413115613231149839938720734032056014045892987014787785672104471033795618564656289493229387484259, 13525270693717786544778294508428969584515672470168042831168877160802165635578464309975799746941160181335288794793191942662579478479357378871004587168098794958399656736754118490991238685510235931767386260189148171, 19235744998060364176844724672149069395018987361032459496049116842353747053803590380652114222297967257760012930056337996550384485709253114398305982204009705188869455282508476735957918585195064453770229686567228938, 20681388393668631721223424741764541346447242888797212955618840837705680426384854075143375026781634326078775533827721666951582003940340520355564490096930846314091669454357750765113131512339164831508152700113070253, 14184898975101817420201564394408211593902495398686833999895293791408008951338668717918959196432955599892396512301918756693774729792115234723130859164872839327103352660364720335419816382409103431898897905535881211], 'aut_phi_ratio': 1296.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 27, 4, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 2, 3, 1], [3, 2, 6, 1], [3, 3, 6, 1], [3, 6, 3, 1], [3, 6, 6, 2], [3, 6, 18, 1], [6, 1, 6, 1], [6, 2, 3, 1], [6, 2, 6, 1], [6, 2, 9, 1], [6, 2, 18, 1], [6, 3, 18, 1], [6, 6, 9, 1], [6, 6, 18, 2], [6, 6, 54, 1], [6, 27, 8, 1], [6, 27, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3.C_6^2.C_3^3.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '144.183', 'autcent_hash': 183, 'autcent_nilpotent': False, 'autcent_order': 144, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_3\\times S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '5832.et', 'autcentquo_hash': 7243380119719690966, 'autcentquo_nilpotent': False, 'autcentquo_order': 5832, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.S_3^3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 27, 4], [3, 1, 2], [3, 2, 12], [3, 3, 6], [3, 6, 33], [6, 1, 6], [6, 2, 36], [6, 3, 18], [6, 6, 99], [6, 27, 32]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '162.40', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3755, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['162.40', 1], ['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 27, 1, 4], [3, 1, 2, 1], [3, 2, 1, 4], [3, 2, 2, 4], [3, 3, 2, 3], [3, 6, 1, 3], [3, 6, 2, 15], [6, 1, 2, 3], [6, 2, 1, 12], [6, 2, 2, 12], [6, 3, 2, 9], [6, 6, 1, 9], [6, 6, 2, 45], [6, 27, 2, 16]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 1456, 'exponent': 6, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '648.754', 'hash': 3755, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 1, 3, 3, 3], 'inner_gens': [[46715, 1166425, 1182673, 1225511, 794033], [46715, 1166425, 78793, 1668518, 793601], [1182167, 1166425, 78793, 1668518, 793601], [502682, 1166425, 78793, 1668518, 793601], [47579, 1166425, 78793, 1668518, 793601]], 'inner_hash': 40, 'inner_nilpotent': False, 'inner_order': 162, 'inner_split': False, 'inner_tex': 'C_3^3:C_6', 'inner_used': [1, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 72], [2, 144], [6, 36]], 'label': '1944.3755', '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': 'C2*C3^3:C6^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 25, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 252, 'number_divisions': 144, 'number_normal_subgroups': 286, 'number_subgroup_autclasses': 285, 'number_subgroup_classes': 2097, 'number_subgroups': 11802, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 111], [3, 242], [6, 1590]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [3, 6, 2, 3, 2, 2, 2, 2, 6, 3], 'outer_gen_pows': [46657, 46657, 47101, 46657, 46657, 46657, 46657, 47101, 46657, 46657], 'outer_gens': [[46715, 1166425, 78361, 1668950, 793601], [327713, 606541, 487141, 478567, 1633859], [887153, 1166425, 1183105, 1668086, 794033], [1167335, 1166425, 486709, 1668086, 793601], [607007, 606541, 89494, 1225943, 793601], [1073941, 606541, 1590589, 1668950, 886915], [1166891, 1166425, 1182241, 1225943, 793601], [233467, 606541, 77929, 1668518, 793601], [514057, 606541, 77929, 1668518, 794033], [607007, 1166425, 78361, 1668950, 793601]], 'outer_group': '5184.qf', 'outer_hash': 6803420753452827215, 'outer_nilpotent': False, 'outer_order': 5184, 'outer_permdeg': 13, 'outer_perms': [348, 43626431, 1037837269, 45630, 5047, 3634284, 5176, 1520472240, 482711699, 80988], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4\\times S_3^3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 48], [4, 64], [6, 12], [12, 12]], 'representations': {'PC': {'code': 1795078616136118212167289476294557716853, 'gens': [1, 3, 4, 5, 7], 'pres': [8, -2, -3, -3, -3, -2, -3, -2, -3, 16, 21891, 10667, 10804, 116, 10373, 90726, 166, 82951]}, 'GLZN': {'d': 2, 'p': 36, 'gens': [1622293, 1166425, 638209, 1632995, 502694, 793169, 46681, 47089]}, 'Perm': {'d': 19, 'gens': [6423296499247855, 6423296495616000, 13516122267648000, 93405312000, 51932, 43644799, 21369, 99199]}}, 'schur_multiplier': [6, 6, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_3^3:C_6^2', 'transitive_degree': 216, 'wreath_data': None, 'wreath_product': False}
