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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '4840.bd', 'ambient_counter': 30, 'ambient_order': 4840, 'ambient_tex': 'C_{44}.F_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 4, 'counter': 18, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4840.bd.22.c1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '22.c1.c1', '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': 22, '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': '220.2', 'subgroup_hash': 2, 'subgroup_order': 220, 'subgroup_tex': 'C_{11}:C_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4840.bd', 'aut_centralizer_order': 4, 'aut_label': '22.c1', 'aut_quo_index': None, 'aut_stab_index': 110, 'aut_weyl_group': '220.7', 'aut_weyl_index': 440, 'centralizer': '1210.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['44.c1.c1', '110.d1.c1', '242.a1.a1'], 'core': '1210.a1.a1', 'coset_action_label': None, 'count': 22, 'diagramx': [5249, -1, 4354, -1, 1822, -1, 1443, -1], 'generators': [1210, 2420, 450, 2422], 'label': '4840.bd.22.c1.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '22.c1.c1', 'old_label': '22.c1.c1', 'projective_image': '2420.y', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '22.c1.c1', 'subgroup_fusion': None, 'weyl_group': '55.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '20.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [10, 22], 'aut_gens': [[1, 5], [1, 65], [201, 115]], 'aut_group': '220.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 220, 'aut_permdeg': 13, 'aut_perms': [143842590, 982342849], 'aut_phi_ratio': 2.75, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 11, 1, 4], [10, 11, 1, 4], [11, 5, 2, 1], [20, 11, 2, 4], [22, 5, 2, 1], [44, 5, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{11}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2], [5, 11, 4], [10, 11, 4], [11, 5, 2], [20, 11, 8], [22, 5, 2], [44, 5, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '55.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 11, 4, 1], [10, 11, 4, 1], [11, 5, 2, 1], [20, 11, 8, 1], [22, 5, 2, 1], [44, 5, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 220, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[5, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '110.2', 'hash': 2, 'hyperelementary': 5, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 55, 'inner_gen_orders': [5, 11], 'inner_gens': [[1, 185], [41, 5]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 55, 'inner_split': True, 'inner_tex': 'C_{11}:C_5', 'inner_used': [1, 2], 'irrC_degree': 5, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 10, 'irrep_stats': [[1, 20], [5, 8]], 'label': '220.2', 'linC_count': 4, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 2, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C11:C20', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_conjugacy_classes': 28, 'number_divisions': 9, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 42, 'old_label': None, 'order': 220, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 44], [10, 44], [11, 10], [20, 88], [22, 10], [44, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [2, 0], 'outer_gens': [[1, 145], [1, 215]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1], [10, 2], [20, 1]], 'representations': {'PC': {'code': 36365577375343009, 'gens': [1, 2], 'pres': [4, -5, -2, -2, -11, 1481, 21, 1802, 34, 1283]}, 'GLFq': {'d': 2, 'q': 121, 'gens': [11317501, 83263414, 165132033, 17715620]}, 'Perm': {'d': 15, 'gens': [9, 7277200680, 16, 100725742080]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [20], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:C_{20}', 'transitive_degree': 44, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [10, 20, 10, 10, 20], 'aut_gens': [[1, 10, 110], [1201, 80, 3410], [4311, 70, 4510], [1731, 40, 1650], [4441, 100, 1430], [3411, 80, 550]], 'aut_group': None, 'aut_hash': 5676471071568951475, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96800, 'aut_permdeg': 165, 'aut_perms': [22847803005773494720042969063247732216221725405688615083198698365608857848829844173122422058121841328833142322919745148925284455597621307113213346827210422569584974513361580183433411837745991785227903445403195969743051390183631455067997100220609649363035855997377437312712647148073661878600583061, 17708696605040736956296763528532302928427618274544511144556790295731402991572784828836842616309313233083976356273269252308237305424803010530278499453262455754509797638989986596034250001799283832580484045691903121824036867255150580702354656514644706840738909216628595357691710049469041318222155411, 5808977891753376748618121891114672312524942501618029352528516461762309051986256243218217189286196419537003502973258529118254847896979782984833482570390261046965200086532059147262359009421722208461650059837172204571330906008275902779927535057633481609843728372101111977940750506174707090663841770, 27962145548170294538757710043243966305247662099255717825072432922304677816271619640798950898308581599264892373525208327818152419314497414866804678252250469471124671787940871705283415512222721436043366944093866913364723218452596248384190741322074625357628946554902075658912524343085241595027291935, 30622069494405944629727463925548557522260405236395746621092981368932652009621369953709660054884384819477200247250612755271151480366560330154687034664945110072092458977824476608540726167913022469166460307263297187642518058464816131156886520814305869199322508882868756060763357426783310407519156339], 'aut_phi_ratio': 55.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 22, 2, 1], [5, 121, 1, 4], [10, 121, 1, 4], [11, 5, 2, 1], [11, 10, 1, 1], [11, 10, 10, 1], [20, 242, 1, 4], [20, 242, 2, 4], [22, 5, 2, 1], [22, 10, 1, 1], [22, 10, 10, 1], [44, 10, 2, 2], [44, 10, 20, 1], [44, 110, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{11}^2.C_{10}^2.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '24200.n', 'autcentquo_hash': 3130384263358372570, 'autcentquo_nilpotent': False, 'autcentquo_order': 24200, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_{11}^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 22, 2], [5, 121, 4], [10, 121, 4], [11, 5, 2], [11, 10, 11], [20, 242, 12], [22, 5, 2], [22, 10, 11], [44, 10, 24], [44, 110, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '2420.y', 'commutator_count': 1, 'commutator_label': '242.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '11.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 30, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 22, 1, 2], [5, 121, 4, 1], [10, 121, 4, 1], [11, 5, 2, 1], [11, 10, 1, 1], [11, 10, 2, 5], [20, 242, 4, 3], [22, 5, 2, 1], [22, 10, 1, 1], [22, 10, 2, 5], [44, 10, 2, 2], [44, 10, 4, 5], [44, 110, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 220, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[10, 0, 20]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2420.y', 'hash': 5002246478290475993, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 11, 22], 'inner_gens': [[1, 60, 2970], [61, 10, 110], [1981, 10, 110]], 'inner_hash': 549729644264252084, 'inner_nilpotent': False, 'inner_order': 2420, 'inner_split': True, 'inner_tex': 'C_{22}:F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 10, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 20, 'irrep_stats': [[1, 20], [2, 5], [5, 8], [10, 46]], 'label': '4840.bd', 'linC_count': 20, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 18, 'linQ_dim': 24, 'linQ_dim_count': 18, 'linR_count': 10, 'linR_degree': 20, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C44.F11', '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': 30, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 79, 'number_divisions': 33, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 46, 'number_subgroup_classes': 78, 'number_subgroups': 1308, 'old_label': None, 'order': 4840, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1], [4, 46], [5, 484], [10, 484], [11, 120], [20, 2904], [22, 120], [44, 680]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 10], 'outer_gen_pows': [2428, 2428, 3630], 'outer_gens': [[2421, 20, 990], [2421, 20, 1430], [1211, 40, 990]], 'outer_group': '40.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 11, 'outer_perms': [3628800, 40320, 753], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 4], [8, 1], [10, 6], [20, 12], [40, 5]], 'representations': {'PC': {'code': '71610021749011032000227273403635942989259358171039009', 'gens': [1, 3, 4], 'pres': [6, -2, -5, -11, -2, -2, -11, 12, 14527, 1082, 278, 71283, 33009, 69, 33004, 9910, 88, 79205, 23771]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [16947971986788040, 11932343999372442, 9033174481858351, 6631581384762157, 41628676495383410, 41772741070013040]}, 'Perm': {'d': 30, 'gens': [654960923937232846399545110371, 9769365937326835238068139606748, 18927338216889315598314214110041, 4830041, 654960923937247114696109666195, 6446835113241600]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{44}.F_{11}', 'transitive_degree': 88, 'wreath_data': None, 'wreath_product': False}
