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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '528.96', 'ambient_counter': 96, 'ambient_order': 528, 'ambient_tex': 'C_{12}.D_{22}', 'central': False, 'central_factor': False, 'centralizer_order': 22, 'characteristic': False, 'core_order': 132, 'counter': 12, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '528.96.4.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': ['C2'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '132.1', 'subgroup_hash': 1, 'subgroup_order': 132, 'subgroup_tex': 'C_3:C_{44}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '528.96', 'aut_centralizer_order': 44, 'aut_label': '4.c1', 'aut_quo_index': 6, 'aut_stab_index': 2, 'aut_weyl_group': '120.45', 'aut_weyl_index': 88, 'centralizer': '24.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1', '2.c1.a1', '2.e1.a1'], 'contains': ['8.a1.a1', '12.d1.a1', '44.b1.a1'], 'core': '4.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3708, 5891, 2459, 5211, 8120, 2185, 7168, 8643], 'generators': [1, 264, 352, 48], 'label': '528.96.4.c1.a1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.c1.a1', 'normal_contained_in': ['2.b1.a1', '2.c1.a1', '2.e1.a1'], 'normal_contains': ['8.a1.a1', '44.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.c1.a1', 'projective_image': '264.34', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.c1.a1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '44.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 30, 'aut_gen_orders': [2, 2, 30], 'aut_gens': [[1, 4], [3, 4], [1, 92], [89, 28]], 'aut_group': '120.45', 'aut_hash': 45, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 120, 'aut_permdeg': 12, 'aut_perms': [40320, 3628800, 43547329], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 3, 2, 1], [6, 2, 1, 1], [11, 1, 10, 1], [22, 1, 10, 1], [33, 2, 10, 1], [44, 3, 20, 1], [66, 2, 10, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{10}\\times D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '20.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 20, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 1], [4, 3, 2], [6, 2, 1], [11, 1, 10], [22, 1, 10], [33, 2, 10], [44, 3, 20], [66, 2, 10]], 'center_label': '22.2', 'center_order': 22, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['12.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 3, 2, 1], [6, 2, 1, 1], [11, 1, 10, 1], [22, 1, 10, 1], [33, 2, 10, 1], [44, 3, 20, 1], [66, 2, 10, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 132, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 0, 10]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '66.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 92], [45, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 20, 'irrQ_dim': 40, 'irrR_degree': 4, 'irrep_stats': [[1, 44], [2, 22]], 'label': '132.1', 'linC_count': 10, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 2, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 15, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:C44', '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 66, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 16, 'old_label': None, 'order': 132, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 6], [6, 2], [11, 10], [22, 10], [33, 20], [44, 60], [66, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 10], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 40], [3, 124]], 'outer_group': '20.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 20, 'outer_permdeg': 9, 'outer_perms': [720, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{10}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [4, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [10, 2], [20, 3]], 'representations': {'PC': {'code': 1606640850019, 'gens': [1, 3], 'pres': [4, -2, -2, -3, -11, 8, 1106, 46]}, 'GLFp': {'d': 2, 'p': 23, 'gens': [166507, 26912]}, 'Perm': {'d': 18, 'gens': [21016674201600, 4037913, 186810624000, 376610217984000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [44], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_{44}', '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': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
