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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1584.67', 'ambient_counter': 67, 'ambient_order': 1584, 'ambient_tex': 'D_{792}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 22, 'counter': 33, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1584.67.36.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '36.b1.b1', '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': 36, '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': '44.3', 'subgroup_hash': 3, 'subgroup_order': 44, 'subgroup_tex': 'D_{22}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1584.67', 'aut_centralizer_order': 24, 'aut_label': '36.b1', 'aut_quo_index': None, 'aut_stab_index': 36, 'aut_weyl_group': '220.7', 'aut_weyl_index': 864, 'centralizer': '792.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.b1.b1', '18.b1.b1'], 'contains': ['72.a1.a1', '72.b1.b1', '396.b1.b1'], 'core': '72.a1.a1', 'coset_action_label': None, 'count': 18, 'diagramx': [5400, -1, 4006, -1, 5298, -1, 5529, -1], 'generators': [1539, 144, 792], 'label': '1584.67.36.b1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.b1.b1', 'old_label': '36.b1.b1', 'projective_image': '792.25', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.b1.b1', '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': '4.2', '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, 2], [1, 26], [19, 2]], 'aut_group': '220.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 220, 'aut_permdeg': 13, 'aut_perms': [87956214, 965432545], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [11, 2, 5, 1], [22, 2, 5, 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], [2, 11, 2], [11, 2, 5], [22, 2, 5]], 'center_label': '2.1', 'center_order': 2, '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', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['22.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [11, 2, 5, 1], [22, 2, 5, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 22, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 11], 'faithful_reps': [[2, 1, 5]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '44.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [2, 11], 'inner_gens': [[1, 42], [5, 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': 10, 'irrQ_dim': 10, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 10]], 'label': '44.3', 'linC_count': 5, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 5, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D22', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 14, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 40, 'old_label': None, 'order': 44, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 23], [11, 10], [22, 10]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [10], 'outer_gen_pows': [0], 'outer_gens': [[23, 38]], 'outer_group': '10.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 10, 'outer_permdeg': 7, 'outer_perms': [810], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{10}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [10, 2]], 'representations': {'PC': {'code': 571694009, 'gens': [1, 2], 'pres': [3, -2, -2, -11, 253, 16, 362]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [1343, 13320, 13311]}, 'Perm': {'d': 13, 'gens': [40284846, 1, 559571286]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{22}', 'transitive_degree': 22, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 3960, 'aut_gen_orders': [18, 120, 180, 30, 30, 18], 'aut_gens': [[1, 2], [1257, 746], [679, 1570], [53, 314], [793, 562], [325, 1270], [731, 878]], 'aut_group': None, 'aut_hash': 2750653646408301733, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 190080, 'aut_permdeg': 792, 'aut_perms': 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'aut_phi_ratio': 396.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 396, 2, 1], [3, 2, 1, 1], [4, 2, 1, 1], [6, 2, 1, 1], [8, 2, 2, 1], [9, 2, 3, 1], [11, 2, 5, 1], [12, 2, 2, 1], [18, 2, 3, 1], [22, 2, 5, 1], [24, 2, 4, 1], [33, 2, 10, 1], [36, 2, 6, 1], [44, 2, 10, 1], [66, 2, 10, 1], [72, 2, 12, 1], [88, 2, 20, 1], [99, 2, 30, 1], [132, 2, 20, 1], [198, 2, 30, 1], [264, 2, 40, 1], [396, 2, 60, 1], [792, 2, 120, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{99}.C_{60}.C_2^5', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1980, 'autcentquo_group': None, 'autcentquo_hash': 7697546556299087385, 'autcentquo_nilpotent': False, 'autcentquo_order': 47520, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{99}.C_{30}.C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 396, 2], [3, 2, 1], [4, 2, 1], [6, 2, 1], [8, 2, 2], [9, 2, 3], [11, 2, 5], [12, 2, 2], [18, 2, 3], [22, 2, 5], [24, 2, 4], [33, 2, 10], [36, 2, 6], [44, 2, 10], [66, 2, 10], [72, 2, 12], [88, 2, 20], [99, 2, 30], [132, 2, 20], [198, 2, 30], [264, 2, 40], [396, 2, 60], [792, 2, 120]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '792.25', 'commutator_count': 1, 'commutator_label': '396.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 67, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 396, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [6, 2, 1, 1], [8, 2, 2, 1], [9, 2, 3, 1], [11, 2, 5, 1], [12, 2, 2, 1], [18, 2, 3, 1], [22, 2, 5, 1], [24, 2, 4, 1], [33, 2, 10, 1], [36, 2, 6, 1], [44, 2, 10, 1], [66, 2, 10, 1], [72, 2, 12, 1], [88, 2, 20, 1], [99, 2, 30, 1], [132, 2, 20, 1], [198, 2, 30, 1], [264, 2, 40, 1], [396, 2, 60, 1], [792, 2, 120, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 792, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 1, 120]], 'familial': True, 'frattini_label': '12.2', 'frattini_quotient': '132.9', 'hash': 67, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 396, 'inner_gen_orders': [2, 396], 'inner_gens': [[1, 1582], [5, 2]], 'inner_hash': 25, 'inner_nilpotent': False, 'inner_order': 792, 'inner_split': False, 'inner_tex': 'D_{396}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 240, 'irrQ_dim': 240, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 395]], 'label': '1584.67', 'linC_count': 120, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 4, 'linQ_dim': 20, 'linQ_dim_count': 4, 'linR_count': 120, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D792', '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': 25, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 399, 'number_divisions': 26, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 66, 'number_subgroups': 2364, 'old_label': None, 'order': 1584, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 793], [3, 2], [4, 2], [6, 2], [8, 4], [9, 6], [11, 10], [12, 4], [18, 6], [22, 10], [24, 8], [33, 20], [36, 12], [44, 20], [66, 20], [72, 24], [88, 40], [99, 60], [132, 40], [198, 60], [264, 80], [396, 120], [792, 240]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 2, 30], 'outer_gen_pows': [1408, 1408, 0, 330], 'outer_gens': [[1409, 1010], [1409, 398], [1, 218], [551, 1202]], 'outer_group': '240.208', 'outer_hash': 208, 'outer_nilpotent': True, 'outer_order': 240, 'outer_permdeg': 16, 'outer_perms': [40279680, 362880, 1307674368000, 6227026593], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{30}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 2], [6, 2], [8, 1], [10, 2], [12, 1], [20, 3], [24, 1], [40, 2], [60, 2], [80, 1], [120, 1], [240, 1]], 'representations': {'PC': {'code': 1202941996938997147046784599807845591982060000119, 'gens': [1, 2], 'pres': [7, -2, -2, -2, -2, -3, -3, -11, 22149, 36, 33182, 58, 44131, 80, 54884, 137, 64517, 166, 70566]}, 'Perm': {'d': 28, 'gens': [868708690480067649339017647, 7891827264921600, 13916445459840000, 21069517179110400, 12145280938112229701713920000, 4364893, 23454917710790789398855680000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{792}', 'transitive_degree': 792, 'wreath_data': None, 'wreath_product': False}
