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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '528.96', 'ambient_counter': 96, 'ambient_order': 528, 'ambient_tex': 'C_{12}.D_{22}', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': True, 'core_order': 264, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '528.96.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '264.14', 'subgroup_hash': 14, 'subgroup_order': 264, 'subgroup_tex': 'C_{12}\\times D_{11}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '528.96', 'aut_centralizer_order': 12, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '880.210', 'aut_weyl_index': 12, 'centralizer': '44.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.b1.a1', '4.d1.a1', '6.a1.a1', '22.b1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6459, 4246, 7169, 778, 6199, 6755, 4415, 1833], 'generators': [2, 352, 264, 132, 48], 'label': '528.96.2.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '4.b1.a1', '4.d1.a1', '6.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '264.34', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.a1.a1', 'subgroup_fusion': None, 'weyl_group': '44.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [2, 2, 10, 22], 'aut_gens': [[1, 2], [133, 178], [1, 46], [1, 146], [241, 178]], 'aut_group': '880.210', 'aut_hash': 210, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 880, 'aut_permdeg': 26, 'aut_perms': [224213760811054763155679511, 156289753242448179782400, 202301637899075251136639215, 86429589077575155209438016], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [3, 1, 2, 1], [4, 1, 2, 1], [4, 11, 2, 1], [6, 1, 2, 1], [6, 11, 4, 1], [11, 2, 5, 1], [12, 1, 4, 1], [12, 11, 4, 1], [22, 2, 5, 1], [33, 2, 10, 1], [44, 2, 10, 1], [66, 2, 10, 1], [132, 2, 20, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3\\times F_{11}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', '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], [2, 11, 2], [3, 1, 2], [4, 1, 2], [4, 11, 2], [6, 1, 2], [6, 11, 4], [11, 2, 5], [12, 1, 4], [12, 11, 4], [22, 2, 5], [33, 2, 10], [44, 2, 10], [66, 2, 10], [132, 2, 20]], 'center_label': '12.2', 'center_order': 12, 'central_product': True, 'central_quotient': '22.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['22.1', 1], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [3, 1, 2, 1], [4, 1, 2, 1], [4, 11, 2, 1], [6, 1, 2, 1], [6, 11, 2, 2], [11, 2, 5, 1], [12, 1, 4, 1], [12, 11, 4, 1], [22, 2, 5, 1], [33, 2, 10, 1], [44, 2, 10, 1], [66, 2, 10, 1], [132, 2, 20, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 132, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 0, 20]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '132.7', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [2, 11], 'inner_gens': [[1, 218], [49, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 22, 'inner_split': True, 'inner_tex': 'D_{11}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 4, 'irrep_stats': [[1, 24], [2, 60]], 'label': '264.14', 'linC_count': 20, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 20, 'linQ_dim': 14, 'linQ_dim_count': 20, 'linR_count': 50, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12*D11', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 16, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 84, 'number_divisions': 18, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 32, 'number_subgroups': 132, 'old_label': None, 'order': 264, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 23], [3, 2], [4, 24], [6, 46], [11, 10], [12, 48], [22, 10], [33, 20], [44, 20], [66, 20], [132, 40]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 10], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[133, 218], [1, 86], [1, 226]], 'outer_group': '40.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 11, 'outer_perms': [720, 3628800, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 2], [10, 2], [20, 3], [40, 1]], 'representations': {'PC': {'code': 4998490599359343846200019, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -3, -11, 2181, 26, 2582, 42, 1603, 78, 6004]}, 'GLFp': {'d': 2, 'p': 109, 'gens': [104557676, 139863133]}, 'Perm': {'d': 18, 'gens': [21010450850664, 1704, 3, 744, 397708329755520]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}\\times D_{11}', 'transitive_degree': 132, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [30, 10, 66, 10, 10, 10], 'aut_gens': [[1, 2, 4], [353, 410, 52], [221, 50, 140], [221, 122, 268], [265, 458, 380], [265, 50, 452], [45, 266, 428]], 'aut_group': None, 'aut_hash': 2794080620950497868, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10560, 'aut_permdeg': 34, 'aut_perms': [266132923286115186864312431077255303131, 169774635102668307700900882797482548932, 230731977899862726653010050998653201964, 290863090188458638265072266033960414778, 73523653425991840549747465417397760303, 288101723972720834301567321276908579853], 'aut_phi_ratio': 66.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 6, 2, 1], [4, 22, 1, 1], [4, 66, 2, 1], [6, 2, 1, 1], [6, 22, 2, 1], [11, 2, 5, 1], [12, 2, 2, 1], [12, 22, 2, 1], [22, 2, 5, 1], [33, 4, 5, 1], [44, 4, 5, 1], [44, 12, 10, 1], [66, 4, 5, 1], [132, 4, 10, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{66}.C_{10}.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '1320.142', 'autcentquo_hash': 142, 'autcentquo_nilpotent': False, 'autcentquo_order': 1320, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 11, 2], [3, 2, 1], [4, 2, 1], [4, 6, 2], [4, 22, 1], [4, 66, 2], [6, 2, 1], [6, 22, 2], [11, 2, 5], [12, 2, 2], [12, 22, 2], [22, 2, 5], [33, 4, 5], [44, 4, 5], [44, 12, 10], [66, 4, 5], [132, 4, 10]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '264.34', 'commutator_count': 1, 'commutator_label': '66.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 96, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['22.1', 1], ['24.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [4, 6, 1, 2], [4, 22, 1, 1], [4, 66, 1, 2], [6, 2, 1, 1], [6, 22, 1, 2], [11, 2, 5, 1], [12, 2, 2, 1], [12, 22, 2, 1], [22, 2, 5, 1], [33, 4, 5, 1], [44, 4, 5, 1], [44, 12, 5, 2], [66, 4, 5, 1], [132, 4, 10, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4032, 'exponent': 132, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, -1, 10]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '264.34', 'hash': 96, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [2, 2, 66], 'inner_gens': [[1, 2, 92], [1, 2, 436], [441, 98, 4]], 'inner_hash': 34, 'inner_nilpotent': False, 'inner_order': 264, 'inner_split': True, 'inner_tex': 'S_3\\times D_{22}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 40, 'irrQ_dim': 80, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 30], [4, 25]], 'label': '528.96', 'linC_count': 90, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 40, 'linQ_dim': 16, 'linQ_dim_count': 32, 'linR_count': 80, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C12.D22', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 19, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 63, 'number_divisions': 24, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 76, 'number_subgroups': 540, 'old_label': None, 'order': 528, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 23], [3, 2], [4, 168], [6, 46], [11, 10], [12, 48], [22, 10], [33, 20], [44, 140], [66, 20], [132, 40]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 10], 'outer_gen_pows': [0, 0, 132], 'outer_gens': [[1, 2, 260], [1, 266, 436], [133, 2, 100]], 'outer_group': '40.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 11, 'outer_perms': [720, 3628800, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 2], [10, 4], [20, 3], [40, 1]], 'representations': {'PC': {'code': 15902190478426614953179616858095069000019, 'gens': [1, 2, 3], 'pres': [6, -2, -2, -2, -2, -3, -11, 1584, 1658, 3932, 50, 4419, 4137, 69, 11044, 2410, 118, 8651]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [12413213, 6612431, 337985, 433456, 287893, 12459677]}, 'Perm': {'d': 22, 'gens': [368047, 56462890706396697600, 109750244591699712000, 163613728419837696000, 6706022400, 4364893]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}.D_{22}', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
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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}
